Question

If x + 1 and x − 1 are the factors of px³ + x² − 2x + q, find the value of p and q.

Answer 1

Let f(x) = px³ + x² − 2x + q

∵ (x + 1) is a factor of f(x)

∴ f (−1) = 0

⇒ p(−1)³ + (−1)² − 2(−1) + q = 0

⇒ −p + 1 + 2 + q = 0

⇒ − p + q + 3 = 0   ...(i)

∵ (x − 1) is a factor of f(x)

∴ f (1) = 0

⇒ p(1)³ + (1)² − 2(1) + q = 0

⇒ p + 1 − 2 + q = 0

⇒ p + q − 1 = 0   ...(ii)

Subtract (i) from (ii),

(p + q − 1) − (q − p + 3) = 0

p + q − 1 − q + p − 3 = 0

2p − 4 = 0

2p = 4

p = `4/2`

p = 2

Now substitute p = 2 in equation (ii)

p + q − 1 = 0

2 + q − 1 = 0

q + 1 =0

q = −1

The values are p = 2 and q = −1.