Question

Find the values of p and q if (x + 1) and (x + 2) are the factors of x³ + px² + 11x + q.

Answer 1

Let f(x) = x³ + px² + 11x + q

∵ (x + 1) is a factor of f(x)

∴ f (−1) = 0

⇒ (−1)³ + p(−1)² + 11(−1) + q

⇒ −1 + p − 11 + q = 0

⇒ p + q − 12 = 0

⇒ p + q = 12   ...(i)

∵ (x + 2) is a factor of f(x)

∴ f (−2) = 0

⇒ (−2)³ + p(−2)² + 11(−2) + q = 0

⇒ −8 + 4p − 22 + q = 0

⇒ 4p + q − 30 = 0

⇒ 4p + q = 30   ...(ii)

Subtract (i) from (ii),

(4p + q) − (p + q) = 30 − 12

3p = 18

p = `18/3`

p = 6

Now substitute p = 6 in equation (i)

6 + q = 12

q = 12 − 6

q = 6

The values are p = 6 and q = 6.