Division of Polynomials

Divide a polynomial by a monomial We can carry out the division term by term. dividend 2 ) 8 ( ¸ + x divisor 2 8 + = x Rough Work x 2 8 + 2 8 + = x x 1 4  Divide the polynomial term by term. 4 + = x The result (quotient) is x + 4 and the remainder is equal to 0. We say that 2x2 + 8x is divisible by 2x.

You may also use the method of long division. Quotient x 2 x 2 8 x 4 + Divisor 2 x Dividend 2 x + 8 x 2 x 8 x 8 x Remainder = 0

Long division (divisor is not a monomial) Consider the long division of ). 1 2 ( ) 3 6 8 + ¸ x Step 1 3 6 8 1 2 x + 4 x 2 8 Step 2 2 8 x + 1 3 6 4 8 2 x (+ 1)(+ 4x) Step 3 4 8 3 6 1 2 x + 4 x + 3 2 + x  Subtract 8x2 + 4x from 8x2 + 6x + 3.

Step 4 4 8 3 6 1 2 x + + 1 x 2 Step 5 2 x 4 8 3 6 + 1 2 x (+ 1)(+ 1) Step 6 4 8 3 6 1 2 x + 1 + Subtract 2x + 1 from 2x + 3. 2

4 8 3 6 1 2 x + divisor ∵ Degree of 2 = 0, degree of 2x + 1 = 1 ∴ Degree of 2 < degree of 2x + 1 ∴ Stop the division process. remainder ∴ Quotient = 4x + 1, remainder = 2 Since the remainder is not equal to 0, we say that 8x2 + 6x + 3 is not divisible by 2x + 1.

Follow-up question Find the quotient and the remainder of (3x2 + 2x3 – 9)  (2x – 1). 1. Rearrange the terms in descending powers of x. 2. Insert the missing term ‘0x’. 2 x 2 + x 1 + 9 3 2 1 - + x 2 3 - x  x2(2x – 1) = 2x3 – x2 4 2 + x - 9 2 4 - x  2x(2x – 1) = 4x2 – 2x 9 2 - x ∴ Quotient = x2 + 2x + 1 Remainder = -8 1 2 - x  1(2x – 1) = 2x – 1 8 -

What is the relation among the dividend, divisor, quotient and remainder? 4 divisor 29 6 dividend 24 29 = 6 4 + 5 remainder 5 In arithmetic, dividend = divisor quotient + remainder

This is known as the division algorithm. In fact, the mentioned relation is also true for polynomials. Consider f(x)  p(x). Let Q(x) and R(x) be the quotient and the remainder respectively. ) ( x Q R f p M dividend = divisor quotient + remainder f(x) = p(x)  Q(x) + R(x) degree of R(x) < degree of p(x) This is known as the division algorithm.

From the above result, we have 8x2 + 6x + 3  (2x + 1)(4x + 1) + 2 e.g. (8x2 + 6x + 3)  (2x + 1) 4 8 3 6 1 2 x + From the above result, we have 8x2 + 6x + 3  (2x + 1)(4x + 1) + 2 dividend divisor quotient remainder

= (x + 3)(2x – 1) + 5 = (x + 3)(2x) – (x + 3)(1) + 5 When a polynomial is divided by x + 3 , the quotient and the remainder are 2x – 1 and 5 respectively. Find the polynomial. By division algorithm, we have the required polynomial = (x + 3)(2x – 1) + 5 = (x + 3)(2x) – (x + 3)(1) + 5 = 2x2 + 6x – x – 3 + 5 = 2x2 + 5x + 2

Follow-up question When 3x2 + 10x – 5 is divided by a polynomial, the quotient and the remainder are x + 4 and 3. Find the polynomial. Let p(x) be the required polynomial. By division algorithm, we have dividend divisor quotient remainder ∴ The required polynomial is 3x – 2.