Websink Prep UKGCSE MathsTextbook contentsMaths textbooksYear 1Year 2Year 4Year 5Year 6GCSEBrowse allPast papersFind a tutorBecome a tutor
GCSE Maths · EdexcelChapter 3 · Fractions
GCSE Maths · Fractions · Higher · Lesson 3.4

GCSE Maths: Simplifying algebraic fractions (Higher)

Part of the GCSE Maths fractions chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

Download worksheet (PDF)Answers (PDF)
3.4

Simplifying algebraic fractions

Edexcel 1MA1: A4, A5
Grade 6–8

An algebraic fraction has algebra in the numerator, denominator or both. The rules are exactly the same as for number fractions: simplify by dividing top and bottom by a common factor.

Key ideas

  • Factorise first. Then cancel factors that appear in both the numerator and the denominator.
  • You can cancel factors (things multiplied), never terms (things added). x + 33 cannot be simplified by cancelling the 3s.
  • Multiply: multiply numerators, multiply denominators, cancelling common factors first.
  • Divide: keep the first fraction, flip the second, then multiply.
Worked example 3.4.1
Simplify 6x²y9xy

Method. To simplify an algebraic fraction, write the top and the bottom as products, then cancel the numbers and each letter that appears on both.

  1. Write the top and the bottom as multiplications. A power like x² means x × x. Writing everything as a product lets us see every factor.
    6x²y = 6 × x × x × y
    9xy = 9 × x × y
  2. Cancel the numbers. The number parts are 6 and 9. Their highest common factor is 3: 6 ÷ 3 = 2 and 9 ÷ 3 = 3.
    69 = 23
  3. Cancel one x. One x on the bottom cancels with one of the two xs on the top, leaving one x on the top.
    x × x on top, x on the bottom
  4. Cancel the y. The one y on the top cancels with the one y on the bottom.
    y on top, y on the bottom
  5. Write what is left. Multiply together everything that has not been crossed out.
    2x3

Check. Try x = 3 and y = 2. Original: top = 6 × 9 × 2 = 108, bottom = 9 × 3 × 2 = 54, so 108 ÷ 54 = 2. Answer: 2 × 3 ÷ 3 = 2. They match. ✓

Watch out. Cancel numbers with numbers and each letter with the same letter. You can only cancel things that are multiplied, never things that are added.

Worked example 3.4.2
Simplify x² + 5xx² − 25

Key words. A factor is something that is multiplied. For example 2 and 3 are factors of 6, and (x + 1) is a factor of 3(x + 1). Factorise means rewrite as a product. Cancel means divide the top and the bottom of a fraction by the same factor. An algebraic fraction is just a fraction with letters in it, and it follows exactly the same rules as number fractions.

Method. Simplify an algebraic fraction: (1) factorise the top, (2) factorise the bottom, (3) cancel the factors that match.

  1. Write the question and say what is wanted. To simplify means: make the top and bottom as small as possible by cancelling every common factor. The answer must have no common factor left.
    x2 + 5xx2 − 25
  2. Factorise the top: x2 + 5x Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    Look for a common factor. A common factor is something that divides into every term.
    The terms are x2, 5x. The biggest number and the highest power of x that divide into all of them is x.
    Take it outside a bracket, and inside the bracket put each term divided by it:
    x2 ÷ x = x
    5x ÷ x = 5
    So x2 + 5x = x(x + 5).
    Check by expanding: x × (x + 5) = x × x, x × 5 → x2 + 5x. ✓
    x2 + 5x = x(x + 5)
  3. Factorise the bottom: x2 − 25 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
    The rule is: a² − b² = (a − b)(a + b).
    Find a and b. x2 = (x)², so a = x. And 25 = 5², so b = 5.
    Put them into the rule: (x − 5)(x + 5).
    Check by expanding: (x − 5)(x + 5) = x2 + 5x − 5x − 25 = x2 − 25. ✓ (the middle terms cancel)
    x2 − 25 = (x − 5)(x + 5)
  4. Write the fraction again using the factorised top and bottom. Now every factor can be seen and compared.
    x(x + 5)(x − 5)(x + 5)
  5. Find the factors that are exactly the same on the top and the bottom, and cancel them. Dividing the top and the bottom by the same factor does not change the value of the fraction. A bracket on the top only cancels with a bracket on the bottom that is exactly the same.
    Matching factors: (x + 5).
    x(x + 5)(x − 5)(x + 5)
    (the crossed-out factors cancel)
  6. Write what is left. This is the answer. Multiply together whatever has not been crossed out.
    x(x − 5)

Check. Choose a number for x that does not make any bottom zero. Try x = 2. The original: top = 14, bottom = -21, so the fraction = -2/3. Our answer at x = 2 = -2/3. They match. ✓

Watch out. You can only cancel factors (things multiplied), never terms (things added). In (x + 6)/x you cannot cancel the x, because the x is added to 6 on top. That is why we always factorise first.

Worked example 3.4.3
Simplify x² + 7x + 10x² − 4

Key words. A factor is something that is multiplied. For example 2 and 3 are factors of 6, and (x + 1) is a factor of 3(x + 1). Factorise means rewrite as a product. Cancel means divide the top and the bottom of a fraction by the same factor. An algebraic fraction is just a fraction with letters in it, and it follows exactly the same rules as number fractions.

Method. Simplify an algebraic fraction: (1) factorise the top, (2) factorise the bottom, (3) cancel the factors that match.

  1. Write the question and say what is wanted. To simplify means: make the top and bottom as small as possible by cancelling every common factor. The answer must have no common factor left.
    x2 + 7x + 10x2 − 4
  2. Factorise the top: x2 + 7x + 10 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 7 and c = 10.
    We need two numbers that multiply to 10 (the last number) and add to 7 (the middle number). List the pairs that multiply to 10:
    1 × 10 = 10,   add: 1 + 10 = 11 ✗
    -1 × -10 = 10,   add: -1 + -10 = -11 ✗
    2 × 5 = 10,   add: 2 + 5 = 7 ✓ this pair works
    Put the pair into two brackets: (x + 2)(x + 5).
    Check by expanding: x² + 5x + 2x + 10 = x2 + 7x + 10. ✓
    x2 + 7x + 10 = (x + 2)(x + 5)
  3. Factorise the bottom: x2 − 4 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
    The rule is: a² − b² = (a − b)(a + b).
    Find a and b. x2 = (x)², so a = x. And 4 = 2², so b = 2.
    Put them into the rule: (x − 2)(x + 2).
    Check by expanding: (x − 2)(x + 2) = x2 + 2x − 2x − 4 = x2 − 4. ✓ (the middle terms cancel)
    x2 − 4 = (x − 2)(x + 2)
  4. Write the fraction again using the factorised top and bottom. Now every factor can be seen and compared.
    (x + 2)(x + 5)(x − 2)(x + 2)
  5. Find the factors that are exactly the same on the top and the bottom, and cancel them. Dividing the top and the bottom by the same factor does not change the value of the fraction. A bracket on the top only cancels with a bracket on the bottom that is exactly the same.
    Matching factors: (x + 2).
    (x + 2)(x + 5)(x − 2)(x + 2)
    (the crossed-out factors cancel)
  6. Write what is left. This is the answer. Multiply together whatever has not been crossed out.
    (x + 5)(x − 2)

Check. Choose a number for x that does not make any bottom zero. Try x = 3. The original: top = 40, bottom = 5, so the fraction = 8. Our answer at x = 3 = 8. They match. ✓

Watch out. You can only cancel factors (things multiplied), never terms (things added). In (x + 6)/x you cannot cancel the x, because the x is added to 6 on top. That is why we always factorise first.

Worked example 3.4.4
Work out x² − 1x + 2 ÷ x + 1x + 2

Key words. A factor is something that is multiplied. For example 2 and 3 are factors of 6, and (x + 1) is a factor of 3(x + 1). Factorise means rewrite as a product. Cancel means divide the top and the bottom of a fraction by the same factor. An algebraic fraction is just a fraction with letters in it, and it follows exactly the same rules as number fractions.

Method. Multiply or divide algebraic fractions: (1) flip the fraction you are dividing by, (2) factorise every top and bottom, (3) write as one fraction, (4) cancel matching factors.

  1. Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal, which is the fraction turned upside down. Here is why it works with numbers: 1/2 ÷ 1/4 asks "how many quarters fit into a half?" The answer is 2, and 1/2 × 4/1 = 2. So: keep the first fraction, change ÷ to ×, and flip the second fraction.
    x2 − 1x + 2 ÷ x + 1x + 2 = x2 − 1x + 2 × x + 2x + 1
  2. Factorise the first top: x2 − 1 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
    The rule is: a² − b² = (a − b)(a + b).
    Find a and b. x2 = (x)², so a = x. And 1 = 1², so b = 1.
    Put them into the rule: (x − 1)(x + 1).
    Check by expanding: (x − 1)(x + 1) = x2 + 1x − 1x − 1 = x2 − 1. ✓ (the middle terms cancel)
    x2 − 1 = (x − 1)(x + 1)
  3. Factorise the first bottom: x + 2 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    x + 2 is a single bracket already. It cannot be factorised any further.
    x + 2 = (x + 2)
  4. Factorise the second bottom: x + 1 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    x + 1 is a single bracket already. It cannot be factorised any further.
    x + 1 = (x + 1)
  5. Write the multiplication again using all the factorised pieces. This lets us see every factor at once.
    (x − 1)(x + 1)(x + 2) × (x + 2)(x + 1)
  6. Write it as a single fraction. To multiply fractions, multiply the tops together and multiply the bottoms together. Do not multiply out the brackets. Keep the factors so that we can cancel them.
    (x − 1)(x + 1) × (x + 2)(x + 2) × (x + 1)
  7. Cancel every factor that appears on both the top and the bottom. A factor on the top cancels with an identical factor on the bottom because we are dividing the top and the bottom by it. The brackets must be exactly the same.
    Matching factors: (x + 1), (x + 2).
    (x − 1)(x + 1)(x + 2)(x + 1)(x + 2)
    (the crossed-out factors cancel)
  8. Write what is left. This is the answer. Multiply together the factors that have not been crossed out.
    (x − 1)

Check. Try x = 2 (it does not make any bottom zero). The original expression gives 1. Our answer gives 1. They match. ✓

Watch out. Only flip the second fraction, never the first. Only cancel matching factors, never terms that are added.

Worked example 3.4.5
Work out x² − 9x + 4 × x + 4x + 3

Key words. A factor is something that is multiplied. For example 2 and 3 are factors of 6, and (x + 1) is a factor of 3(x + 1). Factorise means rewrite as a product. Cancel means divide the top and the bottom of a fraction by the same factor. An algebraic fraction is just a fraction with letters in it, and it follows exactly the same rules as number fractions.

Method. Multiply or divide algebraic fractions: (1) flip the fraction you are dividing by, (2) factorise every top and bottom, (3) write as one fraction, (4) cancel matching factors.

  1. Factorise the first top: x2 − 9 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
    The rule is: a² − b² = (a − b)(a + b).
    Find a and b. x2 = (x)², so a = x. And 9 = 3², so b = 3.
    Put them into the rule: (x − 3)(x + 3).
    Check by expanding: (x − 3)(x + 3) = x2 + 3x − 3x − 9 = x2 − 9. ✓ (the middle terms cancel)
    x2 − 9 = (x − 3)(x + 3)
  2. Factorise the first bottom: x + 4 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    x + 4 is a single bracket already. It cannot be factorised any further.
    x + 4 = (x + 4)
  3. Factorise the second bottom: x + 3 Factorising means writing it as a product (things multiplied together). We need the factors to be visible so that matching ones can cancel.
    x + 3 is a single bracket already. It cannot be factorised any further.
    x + 3 = (x + 3)
  4. Write the multiplication again using all the factorised pieces. This lets us see every factor at once.
    (x − 3)(x + 3)(x + 4) × (x + 4)(x + 3)
  5. Write it as a single fraction. To multiply fractions, multiply the tops together and multiply the bottoms together. Do not multiply out the brackets. Keep the factors so that we can cancel them.
    (x − 3)(x + 3) × (x + 4)(x + 4) × (x + 3)
  6. Cancel every factor that appears on both the top and the bottom. A factor on the top cancels with an identical factor on the bottom because we are dividing the top and the bottom by it. The brackets must be exactly the same.
    Matching factors: (x + 3), (x + 4).
    (x − 3)(x + 3)(x + 4)(x + 3)(x + 4)
    (the crossed-out factors cancel)
  7. Write what is left. This is the answer. Multiply together the factors that have not been crossed out.
    (x − 3)

Check. Try x = 2 (it does not make any bottom zero). The original expression gives -1. Our answer gives -1. They match. ✓

Watch out. Only flip the second fraction, never the first. Only cancel matching factors, never terms that are added.

Watch out

Do not cancel across addition. x + 6x is not equal to 7, and 2x + 42 = x + 2, not 2x + 2. Cancel only after factorising: 2(x + 2) ÷ 2.

Exercise 3D  Simplifying, multiplying and dividing algebraic fractions

Non-calculator
1
FSimplify.
(a)6x9
(b)12x²8x
(c)5x²y15xy²
(d)3x + 69
2
FFactorise and simplify fully.
(a)x² + 4xx + 4
(b)x² − 9x + 3
(c)x² + 5x + 6x + 2
(d)2x² − 8x² − 4x + 4
(e)x² − x − 12x² − 16
(f)3x² + x − 2x² − 1
3
FWork out. Give each answer in its simplest form.
(a)x4 × 8x²
(b)3x5 ÷ 9x10
(c)x + 3x − 2 × x − 2x + 5
(d)x² − 1x + 2 ÷ x + 1x + 2
(e)2x + 4x² − 9 × x − 3x + 2
(f)x² + 3xx² − 1 ÷ xx − 1
4
RRia says x + 6x simplifies to 7. Explain her mistake.
5
PSThe area of a rectangle is (x² + 5x + 6) cm² and its length is (x + 3) cm. Find an expression for its width.
6
RShow that x² − 4x + 2 × xx − 2 simplifies to x.
7
RChallenge. Simplify x² − 5x + 6x² − 4 ÷ x − 3x + 2

Websink Prep UK is an independent site. It is not affiliated with, endorsed by or sponsored by AQA, Pearson Edexcel, OCR, WJEC, Eduqas, CCEA, SQA, GL Assessment, CEM or the Department for Education. All exam board names and trademarks belong to their owners. Past papers are linked from the exam boards' own websites and are not hosted here.

About Websink Prep UK · Find a tutor · Privacy · Terms · Cookie settings