#### Question

The polynomial px^{3} + 4x^{2} – 3x + q is completely divisible by x^{2} – 1; find the values of p and q. Also, for these values of p and q factorize the given polynomial completely.

#### Solution

Let f(x) = px^{3} + 4x^{2} – 3x + q

It is given that f(x) is completely divisible by (x^{2} – 1) = (x + 1)(x – 1).

Therefore, f(1) = 0 and f(-1) = 0

f(1) = p(1)^{3} + 4(1)^{2} – 3(1) + q = 0

p + q + 1 = 0 …(i)

f(-1) = p(-1)^{3} + 4(-1)^{2} – 3(-1) + q = 0

-p + q + 7 = 0 …(ii)

Adding (i) and (ii), we get,

2q + 8 = 0

q = -4

Substituting the value of q in (i), we get,

p = -q – 1 = 4 – 1 = 3

∴ f(x) = 3x^{3} + 4x^{2} – 3x – 4

Given that f(x) is completely divisible by (x^{2} – 1).

Is there an error in this question or solution?

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The Polynomial Px3 + 4x2 – 3x + Q is Completely Divisible by X2 – 1; Find the Values of P and Q. Also, for These Values of P and Q Factorize the Given Polynomial Completely. Concept: Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem.

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