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Question

The expression 4x3bx2+xc 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.

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Solution

P(x) = 4x3bx2+xc
Given (x+1) divides P(x).
So P(-1) = 0
-4 - b - 1 - c = 0
b + c = -5 ---- (1)
Given P(x) gives a remainder of 30 when divided by (2x - 3).
P(32) = 30
4 (323) - b (322) + (32) - c = 30
Simplify to get
9 b + 4 c = - 60 --- (2)
Solving (1) and (2) , we get: c = 3, b = - 8
P(x) = 4x3+8x2+x3
= (x+1) (4 x2+ m x - 3)
Let,
= 4 x3 + (m +4) x2 + x (m -3) - 3
Comparing coefficients, we get: m = 4

Hence: P(x) = (x+1) (4x2 + 4 x - 3) = (x+1) (2x + 3) (2x - 1)


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