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) = 4x³ + 8x + 8x² + 7, g(x) = 2x² – x + 1

Answer 1

We have

f(x) = 4x³ + 8x² + 8x + 7

g(x) = 2x² − x + 1

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)

4x³ + 8x² + 8x + 7 = (2x² - x + 1)(ax + b) + cx + d

4x³ + 8x² + 8x + 7 = 2ax³ - ax² + ax + 2bx² - xb + b + cx + d

4x³ + 8x² + 8x + 7 = 2ax³ - ax² + 2bx² + ax - xb + cx + b + d

4x³ + 8x² + 8x + 7 = 2ax³ + x²(-a + 2b) + x(a - b + c) + b + d

Equating the co-efficient of various Powers of x on both sides, we get

On equating the co-efficient of x³

2a = 4

`a=4/2`

a = 2

On equating the co-efficient of x²

8 = -a + 2b

Substituting a = 2 we get

8 = -2 + 2b

8 + 2 = 2b

10 = 2b

`10/2=b`

5 = b

On equating the co-efficient of x

a - b + c = 8

Substituting a = 2 and b = 5 we get

2 - 5 + c = 8

-3 + c = 8

c = 8 + 3

c = 11

On equating the constant term, we get

b + d = 7

Substituting b = 5, we get

5 + d = 7

d = 7 - 5

d = 2

Therefore, quotient q(x) = ax + b

= 2x + 5

Remainder r(x) = cx + d

= 11x + 2

Hence, the quotient and remainder are q(x) = 2x + 5 and r(x) = 11x + 2