Question

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = 15x³ − 20x² + 13x − 12; g(x) = x² − 2x + 2

Answer 1

Given

f(x) = 15x³ − 20x² + 13x − 12

g(x) = x² − 2x + 2

Here, Degree (f(x)) = 3 and

degree (g(x)) = 2

Therefore, quotient q(x) is of degree 3 - 2 = 1 and Remainder r(x) is of degree less than 2

Let q(x) = ax + b and

r(x) = cx + d

Using division algorithm, we have

f(x) = g(x) x q(x) + r(x)

15x³ − 20x² + 13x − 12 = (x² − 2x + 2)(ax + b) + cx + d

15x³ − 20x² + 13x − 12 = ax³ - 2ax² + 2ax + bx² - 2bx + 2b + cx + d

15x³ − 20x² + 13x − 12 = ax³ - 2ax² + bx² + 2ax - 2bx + cx + 2b + d

15x³ − 20x² + 13x − 12 = ax³ - x²(2a - b) + x(2a - 2b + c) + 2b + d

Equating the co-efficients of various powers of x on both sides, we get

On equating the co-efficient of x³

ax³ = 15x³

a = 15

On equating the co-efficient of x²

2a - b = 20

Substituting a = 15, we get

2(15) - b = 20

30 - b = 20

-b = 20 - 30

-b = -10

b = 10

On equating the co-efficient of x

2a - 2b + c = 13

Substituting a = 15 and b = 10, we get

2(15) - 2(10) + c = 13

30 - 20 + c = 13

10 + c = 13

c = 13 - 10

c = 3

On equating constant term

2b + d = -12

Substituting b = 10, we get

2(10) + d = -12

20 + d = -12

d = -12 - 20

d = -32

Therefore, quotient q(x) = ax+ b

= 15x + 10

Remainder r(x) = cx + d

= 3x - 32

Hence, the quotient and remainder are q(x) = 15x + 10 and r(x) = 3x - 32