Question

If x − 2 is a factor of x³ + px² + qx − 4 and when the expression is divided by x − 3, it leaves a remainder 17, find the values of p and q.

Answer 1

Let f(x) = x³ + px² + qx − 4

∵ (x − 2) is a factor of f(x)

∴ f (2) = 0

⇒ 2³ + p(2)² + q(2) − 4 = 0

⇒ 8 + 4p + 2q − 4 = 0

⇒ 4p + 2q + 4 = 0

Divide by 2,

⇒ 2p + q + 2 = 0

⇒ 2p + q = −2   ...(i)

When f(x) is divided by (x − 3),

Remainder = 17

∴ f (3) = 17

⇒ 3³ + p(3)² + q(3) − 4 = 17

⇒ 27 + 9p + 3q − 4 = 17

⇒ 23 + 9p + 3q = 17

⇒ 9p + 3q = 17 − 23

⇒ 9p + 3q = −6

Divide by 3,

⇒ 3p + q = −2   ...(ii)

Subtract (i) from (ii),

(3p + q) − (2p + q) = (−2) − (−2)

p = 0

Now substitute p = 0 in equation (i)

2p + q = −2

2(0) + q = −2

q = −2

The values are p = 0 and q = −2.