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Mathematics 4024

Quick revision notes · Mathematics 4024 · Chapter 6 of 35

Factorization

Common factors, difference of two squares, quadratics, grouping, and solving by factorising.

6 sectionsWorked examples Common mistakes
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At a glance

Common factors, difference of two squares, quadratics, grouping, and solving by factorising.

  1. Common factors
  2. Difference of two squares
  3. Quadratics x2 + bx + c
  4. Quadratics ax2 + bx + c
  5. Four terms: grouping
  6. Solving by factorising

1Common factors

  • Take out the highest common factor of every term — numbers and letters.
  • 'Factorise fully' means nothing more can be taken out.
  • Check by expanding back.

Worked example

Factorise fully 12x2 y − 18xy2.

  1. HCF of 12 and 18 is 6; of x2 y and xy2 is xy.
  2. 12x2 y ÷ 6xy = 2x and 18xy2 ÷ 6xy = 3y.

6xy(2x − 3y)

2Difference of two squares

  • a2 − b2 = (a − b)(a + b). Both terms must be squares and there must be a minus between them.
  • 16x2 − 25y2 = (4x − 5y)(4x + 5y).
  • Take out a common factor first: 2x2 − 50 = 2(x2 − 25) = 2(x − 5)(x + 5).
  • a2 + b2 does not factorise.

Formulas

a2 − b2 = (a − b)(a + b)

Worked example

Factorise fully 3x2 − 75.

  1. Common factor 3: 3(x2 − 25).
  2. x2 − 25 = (x − 5)(x + 5).

3(x − 5)(x + 5)

3Quadratics x2 + bx + c

  • Find two numbers that multiply to c and add to b.
  • If c is positive, both numbers have the sign of b. If c is negative, the signs are different.
ExpressionNumbersFactorised
x2 + 7x + 123, 4(x + 3)(x + 4)
x2 − 7x + 12−3, −4(x − 3)(x − 4)
x2 + x − 124, −3(x + 4)(x − 3)
x2 − x − 12−4, 3(x − 4)(x + 3)

Worked example

Factorise x2 + 2x − 15.

  1. Multiply to −15, add to 2: 5 and −3.

(x + 5)(x − 3)

4Quadratics ax2 + bx + c

  • Use the ac method: find two numbers that multiply to a × c and add to b.
  • Split the middle term with them, then factorise in pairs.

ac method

  1. Multiply a × c.
  2. Find two numbers with that product and sum b.
  3. Rewrite bx as the two terms.
  4. Factorise the first pair and the second pair.
  5. Take out the common bracket.

Worked example

Factorise 6x2 + x − 2.

  1. ac = −12; numbers 4 and −3.
  2. 6x2 + 4x − 3x − 2.
  3. 2x(3x + 2) − 1(3x + 2).

(2x − 1)(3x + 2)

5Four terms: grouping

  • Group into two pairs, factorise each pair, then take out the common bracket.
  • If the brackets don't match, try grouping the terms differently.

Worked example

Factorise xy − 3x + 2y − 6.

  1. x(y − 3) + 2(y − 3).

(x + 2)(y − 3)

6Solving by factorising

  • Rearrange to = 0, factorise, then set each factor equal to 0.
  • x(x − 5) = 0 gives x = 0 or x = 5 — never divide both sides by x.
  • (2x + 3)(x − 3) = 0 gives x = −3/2 or x = 3.

Worked example

Solve 2x2 − 3x − 9 = 0.

  1. ac = −18: −6 and 3.
  2. 2x2 − 6x + 3x − 9 = 2x(x − 3) + 3(x − 3).
  3. (2x + 3)(x − 3) = 0.

x = 3 or x = −1.5

✗Common mistakes

Don’t writeWrite instead
16x2 − 25 = (4x − 5)2 That expands to 16x2 − 40x + 25. Use (4x − 5)(4x + 5)
6x2 + 9x = 3(2x2 + 3x) Take out 3x as well: 3x(2x + 3)
(x + 5)(x − 3) = 0 → x = 5 or −3 Change the signs: x = −5 or 3
Dividing x2 = 5x by x Rearrange: x(x − 5) = 0, so x = 0 or 5

★In the exam

  • Always check a factorisation by expanding it in your head.
  • If a quadratic will not factorise, the question probably wants the formula.
  • 'Hence solve' means use the factorisation you have just found.