Question

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.

Answer 1

Let f(x) = px³ + 4x² – 3x + q

It is given that f(x) is completely divisible by (x² – 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) = 3x³ + 4x² – 3x – 4

Given that f(x) is completely divisible by (x² – 1)

             3x + 4
`x^2 - 1")"overline(3x^3 + 4x^2 - 3x - 4)`
            3x³             – 3
            –                +              
                       4x²            – 4
                       4x²            – 4
                       –               +    
                                0             
∴ 3x³ + 4x² – 3x – 4 = (x² – 1)(3x + 4)

= (x – 1)(x + 1)(3x + 4)