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Factorising Expressions Guide

Learn how to factorise expressions step by step with common factors, including integer and quadratic expressions. Practice examples included!

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Factorising Expressions Guide

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  1. Match the expressions Some of the expressions below are the same. Match up the ones that are equal then write the others in a way similar to the others. 4(y – 2) 4y – 2y² 3(y + 4) 10 – 5y y(y + 2) 4y – 8 2y² - 4y 2y(y – 2) y² + 2y y(4 – 2y)

  2. Answers 4(y – 2) 4y – 2y² 3(y + 4) 10 – 5y y(y + 2) 4y – 8 5(2 – y) 2y² - 4y 2y(y – 2) y² + 2y y(4 – 2y) 3y + 12

  3. Factorising Expressions Learning outcomes All – To be able to factorise simple expressions with common integer factors Most – To be able to factorise an expression into one pair of brackets Some – To be able to factorise quadratic expressions

  4. An example To factorise an expression we write it using brackets and take out all the common factors. Examples 1. 12a - 16 • Find the highest common factor of the numbers • Look for any common unknown factors • Write the common factors outside the brackets • Write what is left inside the brackets • (Rembering the operation +/-) What is the largest factor of 12 and 16? 4 4 x 3 x a 4 x 4 Common factors? Now add any unknowns So 12a – 16 = 4 ( ) 3a – 4

  5. Example 2 Remember to follow each step. Examples 2. 15ab2 + 10b • Find the highest common factor of the numbers • Look for any common unknown factors • Write the common factors outside the brackets • Write what is left inside the brackets • (Rembering the operation +/-) What is the largest factor of 15 and 10? 5 5 x 3 x a x b x b 5 x 2 x b Common factors? Now add any unknowns So 15ab2 + 10b = 5b ( ) 3ab 2 +

  6. Questions Factorise the following expressions • 3x – 9 • 10 + 4b • 12c – 18c2 • 20xy + 16x2 • 5 – 35x

  7. Task 2 Intermediate GCSE book Page 228 Ex 19.6 Start with Q2

  8. Factorising Quadratics Aim – For students to be able to factorise simple quadratics where the coefficient of x2=1 Level – GCSE grade B

  9. Recap Simplify the expression (x + a)(x + b) (x + a)(x + b) F – First O – Outside I – Inside L – Last Note – use FOIL x × x = x2 x × b = bx a × x = ax a × b = ab x2 + bx + ax + ab = x2 + (a + b)x + ab

  10. So … (x + a)(x + b) = x2 + (a + b)x + ab This is useful when factorising quadratics because… • The coefficient of x is ‘a + b’ • The numberical part is ‘a × b’ Example – Factorise x2 + 7x + 12 You are looking for two numbers a and b s.t. a + b = 7 and ab = 12 1 + 6 = 7 but 1 × 6 = 6 – No good 3 + 4 = 7 and 3 × 4 = 12 – Great! Let a = 3 and b = 4 So x2 + 7x + 12 = (x + 3)(x + 4)

  11. More difficult! Example Factorise x2 – 4x – 5 You are looking for two numbers a and b s.t. a + b = -4 and ab = -5 2 + -6 = -4 but 2 × -6 = -12 – No good 1 + -5 = -4 and 1 × -5 = -5 – Great! Let a = 1 and b = -5 Therefore x2 – 4x – 5 = (x + 1)(x – 5) Note – If their product is negative one must be negative

  12. Task Factorise each of the following expressions • x2 + 4x + 3 • x2 + 8x + 15 • x2 + 9x + 20 • x2 – 3x – 4 • x2 – 7x – 30 • x2 + 4x – 12 • x2 – 5x + 6

  13. Answers • x2 + 4x + 3 = (x + 1)(x + 3) • x2 + 8x + 15 = (x + 3)(x + 5) • x2 + 9x + 20 = (x + 4)(x + 5) • x2 – 3x – 4 = (x – 4)(x + 1) • x2 – 7x – 30 = (x – 10)(x + 3) • x2 + 4x – 12 = (x – 2)(x + 6) • x2 – 5x + 6 = (x – 2)(x – 3)

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