Teach Me Maths

Factorising

How to take out common factors, factorise quadratics and differences of squares, and recognise a repeated bracket.

Factorising reverses expanding

Factorising writes a sum or difference as a product. Always look for a common numerical or algebraic factor first. Fully factorised means that no remaining factor can be broken down further in the required number system.

For a common letter factor, take the lowest power present in every term. Check your answer by expanding it.

Worked example 1

Factorise fully 18x^3y^2-12x^2y^3

  1. The highest common numerical factor is 6. Both terms contain x^2y^2, so take out 6x^2y^2

  2. Your turn. Divide 18x^3y^2 by 6x^2y^2. What remains?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    The first term inside the bracket is 3x

  3. Answer: 6x^2y^2(3x-2y). The second term inside the bracket is -2y

Find two numbers with the right sum and product

To factorise x^2+bx+c, find two numbers with sum b and product c. If the product is negative, their signs are opposite. If the product is positive, their signs agree.

Worked example 2

Factorise x^2+x-20

  1. We need a product of -20 and a sum of 1. The numbers are 5 and -4

  2. Your turn. Check the sum 5+(-4)

    5+(-4)=1

  3. Answer: (x+5)(x-4). The cross terms are 5x-4x=x

Explore factors and roots

Try it: a factor (x-p) gives a root x=p. If the factor is (x+2), the root is -2, not 2. Factorising an expression and solving an equation are different tasks.

Split the middle term

For ax^2+bx+c, find two numbers with product ac and sum b. Split bx using those numbers, then factorise in pairs. The common bracket is a useful check.

Worked example 3

Factorise 6x^2-x-12

  1. The product is 6(-12)=-72 and the sum is -1. Use 8 and -9
  2. Split and group: 6x^2+8x-9x-12=2x(3x+4)-3(3x+4)

  3. Your turn. What is the common bracket?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    The common bracket is 3x+4

  4. Answer: (3x+4)(2x-3)

Worked example 4

Factorise 4a^2-13ab+3b^2

  1. Treat this as a quadratic in a. Split -13ab as -12ab-ab because (-12)(-1)=12
  2. Group: 4a(a-3b)-b(a-3b)

  3. Your turn. After taking out (a-3b), what other factor remains?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    The other factor is 4a-b

  4. Answer: (a-3b)(4a-b)

Look for a difference of squares or a repeated bracket

u^2-v^2=(u-v)(u+v)

This needs a subtraction: a sum of two squares does not factorise this way over the reals. After one factorisation, check whether a factor can be factorised again.

A common factor can be a whole bracket. Keep that bracket together while working out what remains.

Worked example 5

Factorise fully over the integers 3x^4-48

  1. Take out 3: 3(x^4-16). Then use a difference of squares: 3(x^2-4)(x^2+4)

  2. Your turn. Factorise x^2-4

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    x^2-4=(x-2)(x+2)

  3. Answer: 3(x-2)(x+2)(x^2+4). The last factor does not factorise further over the integers.

Worked example 6

Factorise fully 2x(x-3)+5(x-3)-(x-3)^2

  1. Each term contains (x-3). Taking it out gives (x-3)\bigl(2x+5-(x-3)\bigr)

  2. Your turn. Simplify 2x+5-(x-3)

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    2x+5-x+3=x+8

  3. Answer: (x-3)(x+8). The subtraction changes both signs inside the inner bracket.

Common mistakes

  • Leaving a common factor behind. Take out the highest common factor first.
  • Checking the product but not the sum. Your chosen pair must satisfy both conditions.
  • Losing signs when taking out a negative. Divide every term by the negative factor, then expand to check.
  • Stopping too early. Check whether any factor can be factorised further.
  • Giving roots instead of factors. For “factorise”, leave the answer as a product.

Now try it: Factorising practice questions

More on this topic: Factorising worksheet with full solutions