GCSE Maths: Expanding double brackets (Foundation)
Part of the GCSE Maths algebra basics chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Expanding double brackets
An expression like (x + 3)(x + 2) is a product of two brackets. We have already met “a number times a bracket”. Now there are two brackets, and the rule gets one step bigger: every term in the first bracket must be multiplied by every term in the second bracket.
The easiest way to see why is a picture. A rectangle has length (x + 3) and width (x + 2). Cut it into four parts. The area of the whole rectangle is (x + 3)(x + 2), and it is also the total of the four parts.
This is the grid method: put the terms of one bracket along the top and the terms of the other down the side. Each box is the row term times the column term. Then add up all four boxes and collect the like terms. Another way to organise the same four multiplications is F-O-I-L: multiply the First terms, then the Outside terms, then the Inside terms, then the Last terms. Both methods give the same answer. Always count your pieces: two terms times two terms gives four pieces.
Negative terms. Keep the sign with the term, and use the sign rules when multiplying. For (x − 2)(x + 5) the second row of the grid is −2 × x = −2x and −2 × 5 = −10.
Three special cases
Squared brackets. (x + 3)2 means (x + 3)(x + 3), so there are four pieces. The two middle pieces are equal, so you get 2 × 3x: (x + 3)2 = x2 + 6x + 9. It is not x2 + 9.
Difference of two squares. (x + 4)(x − 4) = x2 − 4x + 4x − 16 = x2 − 16. When the two brackets have the same numbers but opposite signs, the middle terms cancel. You will use this again when you factorise.
Brackets with more terms. The rule does not change. Every term in the first bracket multiplies every term in the second: a bracket with 2 terms times a bracket with 3 terms gives 6 pieces.
Key ideas
- Multiply every term in the first bracket by every term in the second, then collect like terms.
- Two terms times two terms makes four pieces (use a grid, or F-O-I-L).
- (x + a)2 = (x + a)(x + a) = x2 + 2ax + a2.
- (x + a)(x − a) = x2 − a2 (difference of two squares).
- Keep signs with their terms. A negative times a negative is positive.
- Higher: for three brackets, expand two, collect, then multiply by the third.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 5 x x2 5x 3 3x 15
Outer: x × 5 = 5x
Inner: 3 × x = 3x
Last: 3 × 5 = 15 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 5x + 3x + 15
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +5x, +3x
add the coefficients: 5 + 3 = 8 so they give 8x
number terms: +15 (only one of this type, so it stays as it is)
Answer: x2 + 8x + 15
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 48 and the answer gives 48. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 7 x x2 7x −2 −2x −14
Outer: x × 7 = 7x
Inner: (−2) × x = −2x
Last: (−2) × 7 = −14 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 7x − 2x − 14
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +7x, −2x
add the coefficients: 7 − 2 = 5 so they give 5x
number terms: −14 (only one of this type, so it stays as it is)
Answer: x2 + 5x − 14
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 10 and the answer gives 10. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(x + 4)2 = (x + 4)(x + 4)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 4 x x2 4x 4 4x 16
Outer: x × 4 = 4x
Inner: 4 × x = 4x
Last: 4 × 4 = 16 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 4x + 4x + 16
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +4x, +4x
add the coefficients: 4 + 4 = 8 so they give 8x
number terms: +16 (only one of this type, so it stays as it is)
Answer: x2 + 8x + 16
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 49 and the answer gives 49. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: 2x × x = 2x2× x −5 2x 2x2 −10x 3 3x −15
Outer: 2x × (−5) = −10x
Inner: 3 × x = 3x
Last: 3 × (−5) = −15 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.2x2 − 10x + 3x − 15
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +2x2 (only one of this type, so it stays as it is)
x terms: −10x, +3x
add the coefficients: −10 + 3 = −7 so they give −7x
number terms: −15 (only one of this type, so it stays as it is)
Answer: 2x2 − 7x − 15
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives −18 and the answer gives −18. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x −6 x x2 −6x 6 6x −36
Outer: x × (−6) = −6x
Inner: 6 × x = 6x
Last: 6 × (−6) = −36 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 − 6x + 6x − 36
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −6x, +6x
add the coefficients: −6 + 6 = 0 so they cancel to give 0
number terms: −36 (only one of this type, so it stays as it is)
Answer: x2 − 36
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives −27 and the answer gives −27. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Watch out
(x + 5)2 is not x2 + 25. Squaring a bracket does not square each term. Write it as two brackets, (x + 5)(x + 5), and expand to get x2 + 10x + 25. Also do not forget the middle term in the answer!Exercise 7E Expanding double brackets
Non-calculatorKey words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 3 x x2 3x 2 2x 6
Outer: x × 3 = 3x
Inner: 2 × x = 2x
Last: 2 × 3 = 6 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 3x + 2x + 6
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +3x, +2x
add the coefficients: 3 + 2 = 5 so they give 5x
number terms: +6 (only one of this type, so it stays as it is)
Answer: x2 + 5x + 6
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 30 and the answer gives 30. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 1 x x2 x 5 5x 5
Outer: x × 1 = x
Inner: 5 × x = 5x
Last: 5 × 1 = 5 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + x + 5x + 5
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +x, +5x
add the coefficients: 1 + 5 = 6 so they give 6x
number terms: +5 (only one of this type, so it stays as it is)
Answer: x2 + 6x + 5
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 32 and the answer gives 32. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 6 x x2 6x 4 4x 24
Outer: x × 6 = 6x
Inner: 4 × x = 4x
Last: 4 × 6 = 24 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 6x + 4x + 24
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +6x, +4x
add the coefficients: 6 + 4 = 10 so they give 10x
number terms: +24 (only one of this type, so it stays as it is)
Answer: x2 + 10x + 24
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 63 and the answer gives 63. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: y × y = y2× y 2 y y2 2y 7 7y 14
Outer: y × 2 = 2y
Inner: 7 × y = 7y
Last: 7 × 2 = 14 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.y2 + 2y + 7y + 14
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.y2 terms: +y2 (only one of this type, so it stays as it is)
y terms: +2y, +7y
add the coefficients: 2 + 7 = 9 so they give 9y
number terms: +14 (only one of this type, so it stays as it is)
Answer: y2 + 9y + 14
Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives 50 and the answer gives 50. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: a × a = a2× a 8 a a2 8a 3 3a 24
Outer: a × 8 = 8a
Inner: 3 × a = 3a
Last: 3 × 8 = 24 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.a2 + 8a + 3a + 24
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.a2 terms: +a2 (only one of this type, so it stays as it is)
a terms: +8a, +3a
add the coefficients: 8 + 3 = 11 so they give 11a
number terms: +24 (only one of this type, so it stays as it is)
Answer: a2 + 11a + 24
Check. Test with numbers. Put a = 3 into the original and into your answer. The original gives 66 and the answer gives 66. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 5 x x2 5x −2 −2x −10
Outer: x × 5 = 5x
Inner: (−2) × x = −2x
Last: (−2) × 5 = −10 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 5x − 2x − 10
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +5x, −2x
add the coefficients: 5 − 2 = 3 so they give 3x
number terms: −10 (only one of this type, so it stays as it is)
Answer: x2 + 3x − 10
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 8 and the answer gives 8. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x −4 x x2 −4x 3 3x −12
Outer: x × (−4) = −4x
Inner: 3 × x = 3x
Last: 3 × (−4) = −12 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 − 4x + 3x − 12
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −4x, +3x
add the coefficients: −4 + 3 = −1 so they give −x
number terms: −12 (only one of this type, so it stays as it is)
Answer: x2 − x − 12
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives −6 and the answer gives −6. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x −6 x x2 −6x −1 −x 6
Outer: x × (−6) = −6x
Inner: (−1) × x = −x
Last: (−1) × (−6) = 6 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 − 6x − x + 6
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −6x, −x
add the coefficients: −6 − 1 = −7 so they give −7x
number terms: +6 (only one of this type, so it stays as it is)
Answer: x2 − 7x + 6
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives −6 and the answer gives −6. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: a × a = a2× a −2 a a2 −2a −5 −5a 10
Outer: a × (−2) = −2a
Inner: (−5) × a = −5a
Last: (−5) × (−2) = 10 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.a2 − 2a − 5a + 10
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.a2 terms: +a2 (only one of this type, so it stays as it is)
a terms: −2a, −5a
add the coefficients: −2 − 5 = −7 so they give −7a
number terms: +10 (only one of this type, so it stays as it is)
Answer: a2 − 7a + 10
Check. Test with numbers. Put a = 3 into the original and into your answer. The original gives −2 and the answer gives −2. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: y × y = y2× y 8 y y2 8y −3 −3y −24
Outer: y × 8 = 8y
Inner: (−3) × y = −3y
Last: (−3) × 8 = −24 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.y2 + 8y − 3y − 24
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.y2 terms: +y2 (only one of this type, so it stays as it is)
y terms: +8y, −3y
add the coefficients: 8 − 3 = 5 so they give 5y
number terms: −24 (only one of this type, so it stays as it is)
Answer: y2 + 5y − 24
Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(x + 2)2 = (x + 2)(x + 2)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 2 x x2 2x 2 2x 4
Outer: x × 2 = 2x
Inner: 2 × x = 2x
Last: 2 × 2 = 4 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 2x + 2x + 4
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +2x, +2x
add the coefficients: 2 + 2 = 4 so they give 4x
number terms: +4 (only one of this type, so it stays as it is)
Answer: x2 + 4x + 4
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 25 and the answer gives 25. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(x − 5)2 = (x − 5)(x − 5)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x −5 x x2 −5x −5 −5x 25
Outer: x × (−5) = −5x
Inner: (−5) × x = −5x
Last: (−5) × (−5) = 25 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 − 5x − 5x + 25
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −5x, −5x
add the coefficients: −5 − 5 = −10 so they give −10x
number terms: +25 (only one of this type, so it stays as it is)
Answer: x2 − 10x + 25
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 4 and the answer gives 4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(x + 10)2 = (x + 10)(x + 10)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 10 x x2 10x 10 10x 100
Outer: x × 10 = 10x
Inner: 10 × x = 10x
Last: 10 × 10 = 100 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 10x + 10x + 100
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +10x, +10x
add the coefficients: 10 + 10 = 20 so they give 20x
number terms: +100 (only one of this type, so it stays as it is)
Answer: x2 + 20x + 100
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 169 and the answer gives 169. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(y − 1)2 = (y − 1)(y − 1)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: y × y = y2× y −1 y y2 −y −1 −y 1
Outer: y × (−1) = −y
Inner: (−1) × y = −y
Last: (−1) × (−1) = 1 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.y2 − y − y + 1
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.y2 terms: +y2 (only one of this type, so it stays as it is)
y terms: −y, −y
add the coefficients: −1 − 1 = −2 so they give −2y
number terms: +1 (only one of this type, so it stays as it is)
Answer: y2 − 2y + 1
Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives 4 and the answer gives 4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: 2x × x = 2x2× x 3 2x 2x2 6x 1 x 3
Outer: 2x × 3 = 6x
Inner: 1 × x = x
Last: 1 × 3 = 3 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.2x2 + 6x + x + 3
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +2x2 (only one of this type, so it stays as it is)
x terms: +6x, +x
add the coefficients: 6 + 1 = 7 so they give 7x
number terms: +3 (only one of this type, so it stays as it is)
Answer: 2x2 + 7x + 3
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 42 and the answer gives 42. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: 3x × x = 3x2× x 4 3x 3x2 12x −2 −2x −8
Outer: 3x × 4 = 12x
Inner: (−2) × x = −2x
Last: (−2) × 4 = −8 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.3x2 + 12x − 2x − 8
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +3x2 (only one of this type, so it stays as it is)
x terms: +12x, −2x
add the coefficients: 12 − 2 = 10 so they give 10x
number terms: −8 (only one of this type, so it stays as it is)
Answer: 3x2 + 10x − 8
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 49 and the answer gives 49. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: 2x × 2x = 4x2× 2x 5 2x 4x2 10x 3 6x 15
Outer: 2x × 5 = 10x
Inner: 3 × 2x = 6x
Last: 3 × 5 = 15 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.4x2 + 10x + 6x + 15
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +4x2 (only one of this type, so it stays as it is)
x terms: +10x, +6x
add the coefficients: 10 + 6 = 16 so they give 16x
number terms: +15 (only one of this type, so it stays as it is)
Answer: 4x2 + 16x + 15
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 99 and the answer gives 99. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: 4x × x = 4x2× x −2 4x 4x2 −8x −1 −x 2
Outer: 4x × (−2) = −8x
Inner: (−1) × x = −x
Last: (−1) × (−2) = 2 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.4x2 − 8x − x + 2
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +4x2 (only one of this type, so it stays as it is)
x terms: −8x, −x
add the coefficients: −8 − 1 = −9 so they give −9x
number terms: +2 (only one of this type, so it stays as it is)
Answer: 4x2 − 9x + 2
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 11 and the answer gives 11. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x −3 x x2 −3x 3 3x −9
Outer: x × (−3) = −3x
Inner: 3 × x = 3x
Last: 3 × (−3) = −9 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 − 3x + 3x − 9
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −3x, +3x
add the coefficients: −3 + 3 = 0 so they cancel to give 0
number terms: −9 (only one of this type, so it stays as it is)
Answer: x2 − 9
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: a × a = a2× a −8 a a2 −8a 8 8a −64
Outer: a × (−8) = −8a
Inner: 8 × a = 8a
Last: 8 × (−8) = −64 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.a2 − 8a + 8a − 64
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.a2 terms: +a2 (only one of this type, so it stays as it is)
a terms: −8a, +8a
add the coefficients: −8 + 8 = 0 so they cancel to give 0
number terms: −64 (only one of this type, so it stays as it is)
Answer: a2 − 64
Check. Test with numbers. Put a = 3 into the original and into your answer. The original gives −55 and the answer gives −55. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: 2x × 2x = 4x2× 2x −5 2x 4x2 −10x 5 10x −25
Outer: 2x × (−5) = −10x
Inner: 5 × 2x = 10x
Last: 5 × (−5) = −25 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.4x2 − 10x + 10x − 25
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +4x2 (only one of this type, so it stays as it is)
x terms: −10x, +10x
add the coefficients: −10 + 10 = 0 so they cancel to give 0
number terms: −25 (only one of this type, so it stays as it is)
Answer: 4x2 − 25
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 11 and the answer gives 11. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: y × y = y2× y 7 y y2 7y −7 −7y −49
Outer: y × 7 = 7y
Inner: (−7) × y = −7y
Last: (−7) × 7 = −49 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.y2 + 7y − 7y − 49
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.y2 terms: +y2 (only one of this type, so it stays as it is)
y terms: +7y, −7y
add the coefficients: 7 − 7 = 0 so they cancel to give 0
number terms: −49 (only one of this type, so it stays as it is)
Answer: y2 − 49
Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives −40 and the answer gives −40. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 2 x x2 2x 5 5x 10
Outer: x × 2 = 2x
Inner: 5 × x = 5x
Last: 5 × 2 = 10 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 2x + 5x + 10
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +2x, +5x
add the coefficients: 2 + 5 = 7 so they give 7x
number terms: +10 (only one of this type, so it stays as it is)
Answer: x2 + 7x + 10
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 40 and the answer gives 40. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(x + 3)2 = (x + 3)(x + 3)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 3 x x2 3x 3 3x 9
Outer: x × 3 = 3x
Inner: 3 × x = 3x
Last: 3 × 3 = 9 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 3x + 3x + 9
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +3x, +3x
add the coefficients: 3 + 3 = 6 so they give 6x
number terms: +9 (only one of this type, so it stays as it is)
Answer: x2 + 6x + 9
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 36 and the answer gives 36. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Key words. A term is one part of an expression, together with the sign in front of it (for example 3x or −5). Like terms have exactly the same letters with exactly the same powers (3x and 5x are like terms, 3x and 5x2 are not). The coefficient is the number in front of the letters.
Method. Collect like terms: group the terms that have the same letters and powers, then add or subtract their coefficients (the numbers in front).
- Write the terms out one by one, each with the sign in front of it. A term is one part of the expression. The sign in front of a term belongs to that term, so in 5x − 2y the terms are +5x and −2y. (A term with no sign in front is positive.)+x, +3, +x, +3, +x, +3, +x, +3
- Sort the terms into groups of like terms. Like terms have the same letters with the same powers, so they are the same kind of thing. We can only add the same kind of thing: 3 apples + 2 apples = 5 apples, but 3 apples + 2 pears cannot be made into one number. Remember that x and x2 are different kinds of term, and so are x and xy.x terms: +x, +x, +x, +x
add the coefficients: 1 + 1 + 1 + 1 = 4 so they give 4x
number terms: +3, +3, +3, +3
add the coefficients: 3 + 3 + 3 + 3 = 12 so they give 12 - Write the answer. Join the results of the groups. Put the term with the highest power first, and the number last.4x + 12
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 24 and the answer gives 24. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. The sign in front of a term goes with it. In 7x − 3x you are taking away 3 lots of x. Also, a letter on its own, such as x, has a coefficient of 1 (and −x has −1).
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x −2 x x2 −2x 7 7x −14
Outer: x × (−2) = −2x
Inner: 7 × x = 7x
Last: 7 × (−2) = −14 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 − 2x + 7x − 14
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −2x, +7x
add the coefficients: −2 + 7 = 5 so they give 5x
number terms: −14 (only one of this type, so it stays as it is)
Answer: x2 + 5x − 14
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 10 and the answer gives 10. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.
Method. Substitute x = 5 into the expanded expression.
- Write the expression. From part (a).x2 + 5x − 14
- Replace x by 5. 52 + 5 × 5 − 14 = 25 + 25 − 14
- Work it out. 50 − 14 = 36 cm2
Check. Use the brackets directly: length = 5 + 7 = 12, width = 5 − 2 = 3, and 12 × 3 = 36. ✓
Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.
Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.
- Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)2 means (a + b)(a + b). It does not mean you can just square each term.(x + 4)2 = (x + 4)(x + 4)
- Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.
First: x × x = x2× x 4 x x2 4x 4 4x 16
Outer: x × 4 = 4x
Inner: 4 × x = 4x
Last: 4 × 4 = 16 - Write down all the pieces. Add together the results of all the multiplications, each with its own sign.x2 + 4x + 4x + 16
- Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +4x, +4x
add the coefficients: 4 + 4 = 8 so they give 8x
number terms: +16 (only one of this type, so it stays as it is)
Answer: x2 + 8x + 16
Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 49 and the answer gives 49. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)
Watch out. Do not write (x + 3)2 = x2 + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x2, 3x, 3x and 9.
Method. Expand each pair of brackets separately, then subtract the answers (taking care with the signs) and collect the like terms.
- Expand the first pair of brackets. Every term in one bracket multiplies every term in the other.(x + 3)(x + 4)
x × x = x2
x × 4 = 4x
3 × x = 3x
3 × 4 = 12
x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +4x, +3x
add the coefficients: 4 + 3 = 7 so they give 7x
number terms: +12 (only one of this type, so it stays as it is)
So it equals x2 + 7x + 12 - Expand the second pair of brackets. Do exactly the same.(x + 1)(x + 6)
x × x = x2
x × 6 = 6x
1 × x = x
1 × 6 = 6
x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: +6x, +x
add the coefficients: 6 + 1 = 7 so they give 7x
number terms: +6 (only one of this type, so it stays as it is)
So it equals x2 + 7x + 6 - Subtract the second answer from the first. Put the second answer in a bracket. The minus sign applies to every term of the second answer, so every sign in it changes.(x2 + 7x + 12) − (x2 + 7x + 6)
- Remove the bracket, changing every sign inside it. Minus a positive term gives a negative term, and minus a negative term gives a positive term.x2 + 7x + 12 − x2 − 7x − 6
- Collect the like terms. The terms in x2 and x cancel.6
Check. Try x = 2 and x = 5 in the original. At x = 2: 6. At x = 5: 6. Both equal 6, as expected. ✓
Watch out. Put the second expanded answer in a bracket before subtracting. Writing − x2 + 7x + 6 forgets to change the signs.
Method. Substitute a number into both expressions. If they give different answers, they are not equal.
- Left side with x = 1. Brackets first, then square.(1 + 5)2 = 62 = 36
- Right side with x = 1. 12 + 25 = 1 + 25 = 26
- Compare. 36 is not 26.So (x + 5)2 ≠ x2 + 25.
- Say what is missing. (x + 5)(x + 5) = x2 + 5x + 5x + 25.The correct answer is x2 + 10x + 25, which at x = 1 is 1 + 10 + 25 = 36. ✓
Check. 36 = 36, so the expansion x2 + 10x + 25 works. ✓
Watch out. The missing middle term is twice the product of the two terms: 2 × x × 5 = 10x.
Method. Each part has area = its height × its width. Add the four areas, then collect like terms.
- Top left part. It is x by x.x × x = x2
- Top right part. It is x high and 4 wide.x × 4 = 4x
- Bottom left part. It is 1 high and x wide.1 × x = x
- Bottom right part. It is 1 by 4.1 × 4 = 4
- Add the four areas and collect like terms. 4x + x = 5x.x2 + 4x + x + 4 = x2 + 5x + 4
Check. Expand (x + 4)(x + 1) directly: x2 + x + 4x + 4 = x2 + 5x + 4. ✓
Method. Expand each pair of brackets separately, then subtract the answers (taking care with the signs) and collect the like terms.
- Expand the first pair of brackets. Every term in one bracket multiplies every term in the other.(x + 5)(x − 5)
x × x = x2
x × (−5) = −5x
5 × x = 5x
5 × (−5) = −25
x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −5x, +5x
add the coefficients: −5 + 5 = 0 so they cancel to give 0
number terms: −25 (only one of this type, so it stays as it is)
So it equals x2 − 25 - Expand the second pair of brackets. Do exactly the same.(x + 3)(x − 3)
x × x = x2
x × (−3) = −3x
3 × x = 3x
3 × (−3) = −9
x2 terms: +x2 (only one of this type, so it stays as it is)
x terms: −3x, +3x
add the coefficients: −3 + 3 = 0 so they cancel to give 0
number terms: −9 (only one of this type, so it stays as it is)
So it equals x2 − 9 - Subtract the second answer from the first. Put the second answer in a bracket. The minus sign applies to every term of the second answer, so every sign in it changes.(x2 − 25) − (x2 − 9)
- Remove the bracket, changing every sign inside it. Minus a positive term gives a negative term, and minus a negative term gives a positive term.x2 − 25 − x2 + 9
- Collect the like terms. The terms in x2 and x cancel.−16
Check. Try x = 2 and x = 5 in the original. At x = 2: −16. At x = 5: −16. Both equal −16, as expected. ✓
Watch out. Put the second expanded answer in a bracket before subtracting. Writing − x2 + 7x + 6 forgets to change the signs.