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Expanding brackets and factoring

Multiply out 3(x + 4) and (x + 3)(x + 5) term by term, and go back the other way by factoring.

Expanding means multiplying out brackets, so that 3(x + 4) becomes 3x + 12. Factoring is the opposite: you take a common number out and put the brackets back, so 3x + 12 becomes 3(x + 4). Each one undoes the other. You already use the idea in mental arithmetic. To work out 7 times 23, you can split 23 into 20 and 3: 7(20 + 3) = 140 + 21 = 161. In algebra, expanding helps you tidy an equation before you solve it, and factoring is one of the main ways to solve harder equations with x² in them.

A number times a bracket

The number outside a bracket multiplies EVERY term inside it. Not just the first one.

Remember

a(b + c) = ab + ac

Multiply the outside number by each term inside, one at a time, and keep each sign.

Example 1

Expand 3(x + 4).

  1. 3 times x is 3x.
  2. 3 times 4 is 12.
  3. Put them together with the sign that was between them: 3x + 12.

Answer: 3x + 12

Example 2

Expand 2(3x − 5).

  1. 2 times 3x is 6x.
  2. 2 times −5 is −10.
  3. Together: 6x − 10.

Answer: 6x − 10

When the number outside is negative, the signs need care. These are the sign rules for multiplying:

MultiplyResultExample
positive and positivepositive2(3) = 6
negative and positivenegative(−2)(3) = −6
negative and negativepositive(−2)(−3) = 6

Example 3

Expand −2(x + 3).

  1. −2 times x is −2x.
  2. −2 times 3 is −6.
  3. Together: −2x − 6.

Answer: −2x − 6

Two brackets

To multiply (x + 3)(x + 5), every term in the first bracket multiplies every term in the second bracket. That makes four products. Then you collect the like terms.

  1. Multiply the first term of the first bracket by each term of the second bracket.
  2. Multiply the second term of the first bracket by each term of the second bracket.
  3. Write down all four results, with their signs.
  4. Collect the like terms. Usually the two middle terms are both x terms.

Example 4

Expand (x + 3)(x + 5).

  1. x times x is x².
  2. x times 5 is 5x.
  3. 3 times x is 3x.
  4. 3 times 5 is 15.
  5. All four: x² + 5x + 3x + 15
  6. Collect the x terms: 5x + 3x = 8x. So it is x² + 8x + 15.

Answer: x² + 8x + 15

Example 5

Expand (x − 2)(x + 6).

  1. x times x is x².
  2. x times 6 is 6x.
  3. −2 times x is −2x.
  4. −2 times 6 is −12.
  5. All four: x² + 6x − 2x − 12
  6. Collect: 6x − 2x = 4x. So it is x² + 4x − 12.

Answer: x² + 4x − 12

A bracket squared means the bracket times itself. So (x + 4)² is (x + 4)(x + 4), and you multiply it out in the same way.

Example 6

Expand (x + 4)².

  1. Write it as two brackets: (x + 4)(x + 4)
  2. x times x is x², and x times 4 is 4x.
  3. 4 times x is 4x, and 4 times 4 is 16.
  4. All four: x² + 4x + 4x + 16
  5. Collect: 4x + 4x = 8x. So it is x² + 8x + 16.
  6. Check with x = 1: (1 + 4)² = 25, and 1 + 8 + 16 = 25.

Answer: x² + 8x + 16

Factoring out a common number

A factor of a number is a number that divides it with nothing left over. To factor 6x + 9, find the biggest number that divides both 6 and 9. That is 3. It is called the greatest common factor. Write it in front of a bracket, and write what is left of each term inside.

Example 7

Factor 6x + 9.

  1. The biggest number that divides 6 and 9 is 3.
  2. 6x divided by 3 is 2x.
  3. 9 divided by 3 is 3.
  4. So 6x + 9 = 3(2x + 3).
  5. Check by expanding: 3(2x) + 3(3) = 6x + 9. Correct.

Answer: 3(2x + 3)

Example 8

Factor 10x − 15.

  1. The biggest number that divides 10 and 15 is 5.
  2. 10x divided by 5 is 2x, and −15 divided by 5 is −3.
  3. So 10x − 15 = 5(2x − 3).
  4. Check: 5(2x) + 5(−3) = 10x − 15. Correct.

Answer: 5(2x − 3)

Factoring x squared expressions

To factor x² + 9x + 20, use the pattern from expanding in reverse. You need two numbers that MULTIPLY to the last number, 20, and ADD to the middle number, 9.

Remember

x² + bx + c = (x + p)(x + q), where p and q multiply to c and add to b.

Example 9

Factor x² + 9x + 20.

  1. List the pairs that multiply to 20: 1 and 20, 2 and 10, 4 and 5.
  2. Add each pair: 1 + 20 = 21, 2 + 10 = 12, 4 + 5 = 9.
  3. The pair 4 and 5 adds to 9.
  4. So x² + 9x + 20 = (x + 4)(x + 5).
  5. Check: x² + 5x + 4x + 20 = x² + 9x + 20. Correct.

Answer: (x + 4)(x + 5)

When the last number is negative, the two numbers have different signs: one is positive and one is negative.

Example 10

Factor x² − x − 6.

  1. We need two numbers that multiply to −6 and add to −1 (the middle term −x is −1x).
  2. Pairs that multiply to −6: 1 and −6 (add to −5), −1 and 6 (add to 5), 2 and −3 (add to −1), −2 and 3 (add to 1).
  3. The pair 2 and −3 adds to −1.
  4. So x² − x − 6 = (x + 2)(x − 3).
  5. Check: x² − 3x + 2x − 6 = x² − x − 6. Correct.

Answer: (x + 2)(x − 3), which is the same as (x − 3)(x + 2)

Try it yourself

  1. 1.Expand 6(x − 3).

    Show the answer

    6x − 18

    6 times x is 6x, and 6 times −3 is −18.

  2. 2.Expand −3(2x − 4).

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    −6x + 12

    −3 times 2x is −6x. −3 times −4 is +12, because a negative times a negative is positive.

  3. 3.Expand (x + 2)(x + 7).

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    x² + 9x + 14

    The four products are x² + 7x + 2x + 14. Collect 7x + 2x = 9x.

  4. 4.Expand (x − 5)².

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    x² − 10x + 25

    (x − 5)(x − 5) = x² − 5x − 5x + 25. The last term is +25, because (−5)(−5) = 25.

  5. 5.Factor 8x + 20.

    Show the answer

    4(2x + 5)

    4 is the biggest number that divides 8 and 20. 8x divided by 4 is 2x, and 20 divided by 4 is 5.

  6. 6.Factor x² + 2x − 15.

    Show the answer

    (x + 5)(x − 3)

    5 and −3 multiply to −15 and add to 2.

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