AS Mathematics Algebra Long division Objectives Be able

AS Mathematics Algebra – Long division

Objectives • Be able to divide polynomials using algebraic long division • Be able to factorise polynomials up to order 3 given one linear factor

Long Division Let’s start by looking at a numerical example! We need to set it out the ‘old Example 1 fashioned’ way Divide 364 by 13 13 goes into 36 twice 28 remainder 10 (36 – 26 = 10) 13 364 …bring down the 4 - 26 10 4 13 goes into 104 8 times - 104 remainder 0 (104 – 104 = 0) 0 ANSWER: 28

Example 2 572 divided by 16 35 16 572 - 48 92 - 80 12 16 goes into 57 3 times 3 x 16 = 48 remainder 9 (57 – 48 = 10) …bring down the 2 16 goes into 92 5 times 5 x 16 = 80 remainder 12 (92 – 80 = 12) ANSWER: 3512/16 or 35 ¾

Let’s try with algebra! Example 3 Look at theatfirst the term second in both termexpressions in both expressions What is x 2 ÷ x ? Find (x 2 + x - 2) ÷ (x + 1) x +2 x - 1 x 2 + x - 2 - x 2 x - 2 0 x goes into x 2, x times x(x – 1) = x 2 - x remainder 2 x …bring down the - 2 x goes into 2 x twice 2(x – 1) = 2 x - 2 remainder 0

So (x 2 + x - 2) ÷ (x - 1) = x + 2 Notice quadratic ÷ linear => linear What if we multiply (x - 1) by (x + 2) ? (x - 1)(x + 2) = x 2 + x - 2 We found that (x – 1) goes into x 2 + x – 2 exactly (x + 2) times. . . (x – 1) and (x + 2) are the factors of x 2 + x – 2

Example 4 the term second termexpressions in both Look at theatfirst in both expressions 3 ÷ x ? What is x Divide the cubic expression x goes into x 3, x 2 times x 3 + 2 x 2 - 5 x - 6 by x + 3 x 2(x + 3) = x 3 + 3 x 2 - 2 x remainder - x 2 x + 3 x 3 + 2 x 2 - 5 x - 6 …bring down the - 5 x - (x 3 + 3 x 2) x goes into -x 2, - x times - x 2 - 5 x - x(x + 3) = - x 2 - 3 x remainder - 2 x - -2 x 6 …bring down the - 6 (- 2 x - 6) x goes into -2 x, - 2 times 0 - 2(x + 3) = - 2 x - 6 remainder 0

So (x 3 + 2 x 2 - 5 x - 6) ÷ (x + 3) = x 2 - x - 2 Notice cubic ÷ linear => quadratic x 3 + 2 x 2 - 5 x - 6 (x + 3)(x 2 - x - 2) (x + 3) and (x 2 – x – 2) are factors of x 3 + 2 x 2 - 5 x – 2 BUT (x 2 - x - 2) can be factorised. (x 2 - x - 2) = (x + 1)(x - 2) So x 3 + 2 x 2 - 5 x - 6 = (x + 3)(x + 1)(x - 2) x 3 + 2 x 2 - 5 x – 2 has 3 linear factors (x + 3), (x + 1) and (x - 2)

Questions 1. Divide: a) 2 x 3 - 3 x 2 - 3 x + 2 by x-2 b) 2 x 3 + 3 x 2 - 1 by 2 x - 1 c) 2 x 3 - 11 x 2 + 12 x - 35 by x-5 d) 2. Factorise the above expressions completely

Answers 1. a) 2 x 2 + x - 1 b) x 2 + 2 x + 1 c) 2 x 2 - x + 7 2. a) (x - 2)(2 x 2 + x - 1) = (x - 2)(2 x - 1)(x + 1) b) (2 x - 1)(x 2 + 2 x + 1) = (2 x - 1)(x + 1)2 c) (x - 5)(2 x 2 – x + 7)