Question

find the value of p and q such that (x-1) and (x+2) are factors of $x^{3} + 10x^{2} + px + q$ and factories completely

Answer 1

Answer:

p = 7; q = - 18; (x - 1)(x + 2)(x + 9)

Step-by-step explanation:

If (x - 1) is factor of f(x) = x³ + 10x² + px + q, then

f(1) = 1³ + 10(1)² + p(1)+ q = 0 ⇒ p + q = - 11

If (x + 2) is factor of f(x) = x³ + 10x² + px + q, then

f(- 2) = (- 2)³ + 10(- 2)² + p(- 2) + q = 0

- 8 + 40 - 2p + q = 0 ⇒ q - 2p = - 32

$\left \{ {{q + p = -11} \atop {q - 2p = - 32}} \right.$

(1) - (2)

3p = 21 ⇒ p = 7

q + 7 = - 11 ⇒ q = - 18

f(x) = x³ + 10x² + 7x - 18

(x - 1)(x + 2) = x² + x - 2

(x³ + 10x² + 7x - 18) ÷ (x² + x - 2) = (x + 9)

f(x) = x³ + 10x² + 7x - 18 = (x - 1)(x + 2)(x + 9)