FACTORISING 2

REMINDER:

Factorising expressions of the form x2 + bx + c Factorising is the reverse of expanding. EXPAND 3 + 7 3 × 7 FACTORISE To factorise you need to find two numbers that multiply to give the constant and add to give the coefficient of x.

1 Factorise x2 + 7x + 10. List the pairs of numbers that MULTIPLY to give 10 1 and 10 −1 and −10 2 and 5 −2 and −5 and check to see if they add to give 7. check: (x + 2)(x + 5) = x2 + 5x + 2x + 10 = x2 + 7x + 10

2 Factorise x2 + 9x + 14. List the pairs of numbers that MULTIPLY to give 14 1 and 14 −1 and −14 2 and 7 −2 and −7 and check to see if they add to give 9. check: (x + 2)(x + 7) = x2 + 7x + 2x + 14 = x2 + 9x + 14

3 Factorise x2 − 3x − 10. List the pairs of numbers that MULTIPLY to give −10 1 and −10 −1 and 10 2 and −5 −2 and 5 and check to see if they ADD to give −3. check: (x + 2)(x − 5) = x2 − 5x + 2x − 10 = x2 − 3x − 10

4 Factorise x2 − 8x + 15. List the pairs of numbers that MULTIPLY to give 15 1 and 15 −1 and −15 3 and 5 −3 and −5 and check to see if they ADD to give −8. check: (x − 3)(x − 5) = x2 − 5x − 3x + 15 = x2 − 8x + 15

Simplifying fractions To simplify fractions you must first factorise and then cancel the factors.

1 Simplify factorise divide numerator and denominator by (x + 6)

2 Simplify factorise divide numerator and denominator by (x + 9)

3 Simplify factorise divide numerator and denominator by (x + 4)

The difference of two squares EXPAND FACTORISE In general x2 – a2 = (x + a) (x – a) This result is called the difference of two squares.

1 Factorise x2 − 49. check: (x + 7)(x − 7) = x2 − 7x + 7x − 49 = x2 − 49

2 Factorise 4x2 − 1. check: (2x + 1)(2x − 1) = 4x2 − 2x + 2x − 1 = 4x2 − 1

2 Factorise 16x2 − 25. check: (4x + 5)(4x − 5) = 16x2 − 20x + 20x − 25 = 16x2 − 25

Factorising expressions of the form ax2 + bx + c To factorise 2x2 + 7x + 3 first look at the x2 term, 2x2 must come from 2x × x, so use (2x + ?)(x + ?) next look at the constant term, 3 must come from 3 × 1 or 1 × 3 try expanding (2x + 3)(x + 1) and (2x + 1)(x + 3) to find which is correct So

1 Factorise 5x2 come from 5x × x, so use (5x + ?)(x + ?) −8 comes from 1 × −8 or −1 × 8 or 2 × −4 or −2 × 4 So

2 Simplify factorise divide numerator and denominator by (x − 2)

3 Simplify factorise divide numerator and denominator by (3x + 1)