Question

(Right Answer Please, It's a Humble Request)
Find the value of k if the polynomial p(x)=3x³+kx²+5x-16 is divided by x-2 leaves a remainder -8

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomial is

$\rm :\longmapsto\:p(x) = {3x}^{3} + {kx}^{2} + 5x - 16$

Since, it is given that p(x) leaves the remainder- 8, when divided by x - 2.

We know, By Remainder Theorem,

If a polynomial f(x) is divided by linear polynomial x- a, it leaves the remainder f(a).

So, by using remainder theorem,

$\rm :\longmapsto\:p(2) = - \: 8$

$\rm :\longmapsto\:{3(2)}^{3} + {k(2)}^{2} + 5(2) - 16 = - \: 8$

$\rm :\longmapsto\:24 + 4k + 10 - 16 = - 8$

$\rm :\longmapsto\:18 + 4k = - 8$

$\rm :\longmapsto\: 4k = - 8 - 18$

$\rm :\longmapsto\: 4k = - 26$

$\rm \implies\:k = - \: \dfrac{13}{2}$