Year 11 Extended · Unit 4 Rational expressions and functions · Lesson 1 of 7

When can a factor be cancelled, and what does cancelling hide?

An algebraic fraction can be written in a shorter form by factorizing the top and the bottom and cancelling a common factor. This lesson shows why only factors can be cancelled, and which values of x the fraction can never take.

FunctionsNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: factorize x2 + 7x + 12.
    Answer(x + 3)(x + 4).
  2. From Year 9: factorize x2 − 49.
    Answer(x − 7)(x + 7), a difference of two squares.
  3. From Year 7: simplify 18⁄30.
    Answer3⁄5. Divide top and bottom by 6.
Words used in this lesson
algebraic fraction
a fraction with letters in the top, the bottom or both. For example: (x + 2) / (x + 4).
factor
a part of a product, such as a bracket. For example: x − 4 in (x − 4)(x + 2).
cancel
divide top and bottom by the same factor. For example: 6x / 9 becomes 2x / 3.
excluded value
a value of x that makes the denominator zero. For example: x = 2 for 1 / (x − 2).
denominator
the bottom of a fraction. For example: x2 − 16.

What does cancelling really do?

Esi is checking a fraction from a surveyor's formula: (x2 − 2x − 8) ÷ (x2 − 16). She factorizes both parts, cancels x − 4, and gets (x + 2) ÷ (x + 4).

Predict first: will the two fractions agree for every value of x? Think about a value that makes a factor of the original zero. Then slide x along the line in the figure and watch where the two bars differ.

0
0.5original fraction
0.5cancelled form

x = 0: the original fraction gives 0.5 and the cancelled form gives 0.5.

Show me the steps
  1. x = 0: the original fraction is −8 ÷ −16 = 0.5 and the cancelled form is 2 ÷ 4 = 0.5. They agree.
  2. x = 4: the original has denominator 16 − 16 = 0, so it has no value, but the cancelled form gives 6 ÷ 8 = 0.75. Cancelling has hidden the value x = 4.
  3. x = −4: both denominators are 0, so neither has a value. The excluded values are x = 4 and x = −4, read from the original.

Factorize, then cancel a common factor. A fraction is simplified by writing the top and the bottom as products and cancelling any factor that appears in both. Only factors can be cancelled, never terms that are added or subtracted.

The original fraction cannot take any value of x that makes its denominator zero. Those excluded values are read from the original denominator, before cancelling, and they still apply afterwards.

Terms are not factors. In (x + 5) ÷ 5 the 5s are added to x on the top, so they cannot be cancelled. Test it: at x = 10 the fraction is 15 ÷ 5 = 3, but the “cancelled” answer x would give 10.

Worked examples

1. Simplify (6x + 18) ÷ (9x).

  1. Factorize: 6x + 18 = 6(x + 3) and 9x = 9 × x.
  2. Cancel the common factor 3: 2(x + 3) ÷ (3x), with x ≠ 0.

2. Simplify (x2 + 5x + 6) ÷ (x2 − 9), and state the values x cannot take.

  1. Factorize: (x + 2)(x + 3) on the top and (x − 3)(x + 3) on the bottom.
  2. Cancel x + 3: (x + 2) ÷ (x − 3). The original denominator is zero at x = 3 and at x = −3, so x ≠ 3 and x ≠ −3.

3. Simplify (2x2 − 8) ÷ (x2 − 4x + 4).

  1. Take out 2 first: 2(x2 − 4) = 2(x − 2)(x + 2). The bottom is (x − 2)(x − 2).
  2. Cancel one x − 2: 2(x + 2) ÷ (x − 2), with x ≠ 2.

Your go. The first step is given. Simplify (x2 − 25) ÷ (x2 + 2x − 15). The top is (x − 5)(x + 5).

  1. Factorize the bottom: two numbers with product −15 and sum 2.
  2. Cancel the common factor and state the result.
Check the answer

The bottom is (x + 5)(x − 3), so the result is (x − 5) ÷ (x − 3), with x ≠ −5 and x ≠ 3.

Check before you go on

1. Which is the simplest form of (x2 − 6x) ÷ (x2 − 36)?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. (4x + 20) ÷ (x2 − 25) simplifies to p ÷ (x − q). Find p + q.

3. Simplify (x2 + 3x − 10) ÷ (x2 − 4), then evaluate the result at x = 8. Give your answer as a decimal.

Level 2 · Building

4. Find the sum of the two values of x that the fraction (x − 3) ÷ (x2 − 2x − 15) cannot take.

5. (2x2 − 18) ÷ (x2 + 6x + 9) simplifies to a(x − p) ÷ (x + q). Find a + p + q.

Level 3 · Stretch

6. (x2 + kx − 12) ÷ (x2 − 9) simplifies to (x + 4) ÷ (x + 3). Find k.

7. A vegetable plot in Mombasa has area (x2 − 5x − 24) square metres and length (x + 3) metres. Its width is 11 metres. Find its perimeter in metres.

Something to think about

Esi says: “Cancelling made the fraction simpler, so the excluded value x = 4 no longer matters.” Is she right? Where would you see the difference on a graph?

One way to think about it

No. The original fraction still has no value at x = 4, so the excluded value stays with it. On a graph of the original there is a gap, a missing point, at x = 4, even though the cancelled form is a smooth line there.

Exit ticket

8. For which values of x is (x + 5) ÷ (x2 − 5x) not defined?