Find the values of a and b so that the polynomial (x⁴ + ax³ – 7x² - 8x + b) is exactly divisible by (x + 2) as well as (x + 3).

Question

Find the values of a and b so that the polynomial (x 4 + ax 3 – 7x 2 - 8x + b) is exactly divisible by (x + 2) as well as (x + 3).

Philoid · Accepted Answer

Given, x⁴ + ax³ – 7x² - 8x + b = 0

⸫ x = -2, -3 are a root of the above equation (⸪ they are exactly divisible)

Substituting the value -2 and -3 in place of x will give,

⇒ (-2)⁴ + a (-2)³– 7(-2)² - 8(-2) + b = 0

⇒ 16 – 8a – 28 + 16 + b = 0

⸫ 8a – b = 4 …. (i)

⇒ (-3)⁴ + a (-3)³– 7(-3)² - 8(-3) + b = 0

⇒ 81 – 27a – 63 + 24 + b = 0

⸫ 27a – b = 42 …. (ii)

Simultaneously solving eq(i) and eq(ii) we get,

⸫ a = 2

⸫ b = 12

Find the values of a and b so that the polynomial (x4 + ax3 – 7x2 - 8x + b) is exactly divisible by (x + 2) as well as (x + 3).