Question

The expression $4x^3-b{x}^2+x-c$ leaves remainders 0 and 30 when divided by x+1 and 2x-3 respectively. Calculate the values of b and c hence, factorize the expression completely.

Answer 1

P(x) = $4x^3-b{x}^2+x-c$
Given (x+1) divides P(x).
So P(-1) = 0
$\Rightarrow$ -4 - b - 1 - c = 0
$\Rightarrow$ b + c = -5 ---- (1)
Given P(x) gives a remainder of 30 when divided by (2x - 3).
P($\frac{3}{2}$) = 30
$\Rightarrow$ 4 ${(\frac{3}{2}}^3)$ - b ${(\frac{3}{2}}^2)$ + ($\frac{3}{2}$) - c = 30
Simplify to get
9 b + 4 c = - 60 --- (2)
Solving (1) and (2) , we get: c = 3, b = - 8
P(x) = $4x^3+8x^2+x-3$
= (x+1) (4 $x^2$+ m x - 3)
Let,
= 4 $x^3$ + (m +4) $x^2$ + x (m -3) - 3
Comparing coefficients, we get: m = 4

Hence: P(x) = (x+1) (4$x^2$ + 4 x - 3) = (x+1) (2x + 3) (2x - 1)