Question

The remainder when p\left(x\right)=x^3-2x^2+8x+kp ( x ) = x 3 − 2 x 2 + 8 x + k by (x-2) is 19. What is the remainder when p(x) is divided by (x+2)?

Answer 1

Answer:

$k=3$

The remainder of $p(x)$ divided by $x+2$ is $-29$.

Step-by-step explanation:

The remainder of a polynomial, $p(x)$, divided by $x-a$, $p(a)$.

So the remainder of $p(x)=x^3-2x^2+8x+k$ divided by $x-2$ is $p(2)$ which is given as $19$.

We therefore need to solve $x^3-2x^2+8x|_\text{x=2}=19$ for $k$.

$2^3-2(2)^2+8(2)+k=19$

$8-8+16+k=19$

$16+k=19$

Subtract 16 on both sides:

$k=19-16=3$

So the polynomial is $p(x)=x^3-2x^2+8x+3$.

Now what is the remainder when $p(x)=x^3-2x^2+8x+3$ is divided by $x+2$. To find that remainder we must evaluate $p(-2)$.

$p(-2)=(-2)^3-2(-2)^2+8(-2)+3$

$p(-2)=-8-8-16+3$

$p(-2)=-16-16+3$

$p(-2)=-32+3$

$p(-2)=-29$