Question

find the value of k if (2x^3 + x^2 + k) has a remainder (-9) when divided by (x + 2).

Explanation:

Step1: Apply Remainder Theorem

If polynomial $P(x)$ is divided by $x-a$, remainder is $P(a)$. Here divisor is $x+2=x-(-2)$, so evaluate $P(-2)=-9$.

Step2: Substitute $x=-2$ into polynomial

$P(-2)=2(-2)^3 + (-2)^2 + k = -9$

Step3: Calculate the known terms

$2(-8) + 4 + k = -9$
$\implies -16 + 4 + k = -9$
$\implies -12 + k = -9$

Step4: Solve for $k$

$k = -9 + 12$
$\implies k=3$

Answer:

3