Question

Find the values of "a" and "b" given that p(x)=(x²+3x+2)(x²+4x+a) g(x)=(x²-6x+9)(x²+4x+b) and their G.C.D is (x+2)(x-3)

Answer 1

answer : a = -21 and b = -12

given, P(x) = (x² + 3x + 2)(x² + 4x + a)

g(x) = (x² - 6x + 9)(x² + 4x + b)

and their G.C.D = (x + 2)(x - 3)

P(x) = (x² + 2x + x + 2)(x² + 4x + a)

= (x + 2)(x + 1)(x² + 4x + a)

(x + 2)(x - 3) is a factor of P(x),

so, x = 3 is a zero of P(x).

P(3) = (3 + 2)(3 + 1)(3² + 4 × 3 + a) = 0

⇒5 × 4 × (9 + 12 + a) = 0

⇒a = -21

g(x) = (x² - 2.3.x + 3²)(x² + 4x + b)

= (x - 3)²(x² + 4x + b)

(x + 2)(x - 3) is a factor of g(x).

so, x = 2 is a zero of polynomial g(x).

g(2) = (2 - 3)²(2² + 4 × 2 + b) = 0

⇒(4 + 8 + b) = 0

⇒b = -12