The expression 8x^3-4x^2+2px-q leaves a remainder of -19

Question

when divided by x+1 and a remainder of 2 when divided by 2x-1 . find p and q?

Answer 1

The polynomial function f(x) = 8x³- 4x² + 2px - q

From Remainder theorem,

When we divide a polynomial f(x) by x - c the remainder r equals f(c).

One divisor x + 1

f(- 1) = 8(- 1)³- 4(- 1)² + 2p(- 1) - q

- 19 = - 8 - 4 - 2p + q

- 19 = - 12 - 2p + q

- 7 = - 2p + q

2p - q = 7 ---> (1)

 

And other divisor 2x - 1

f(1/2) = 8(1/2)³ - 4(1/2)² + 2p(1/2) - q

2 = 1 - 1 + p - q

2 = p - q ---> (2)

 

Solve the equations (1) and (2).

From (2), p = 2 + q

Substitute p value in equation (1).

2(2 + q) - q = 7

4 + 2q - q =  7

q = 7 - 4

q = 3

Substitute p value in equation (2).

2 = p - 3

p = 2 + 3

p = 5

The values : p = 5 and q = 3.