Question

The polynomials p (x)= ax^3+3x^2-3 and q (x) = 2x^3 -5x + a when divided by (x-4) leave the remainder r1+ r2=0. Find a

Answer 1

Given that r₁ and r₂ are the remainder of The polynomials p (x)= ax^3+3x^2-3 and q (x) = 2x^3 -5x + a when divided by (x-4).

So using remainder theorem we know that

when  p (x)= ax^3+3x^2-3 divided by x-4, so

r₁ = p(4)

⇒ r₁ = a(4)³+3(4)²-3

⇒ r₁ = 64a+48-3

⇒ r₁ = 64a+45          ...(1)

when  q (x) = 2x^3 -5x + a divided by x-4, so

r₂ = q(4)

⇒ r₂ = 2(4)³-5(4)+a

⇒ r₂ = 128-20+a

⇒ r₂ = 108+a               ...(2)

Given that r₁+r₂ = 0

64a+45+108+a = 0         [ using (1) and (2) ]

⇒ 65a = -153

⇒ a = -153/65