Question

what is the remainder of $(x^3 - 5x^2 - 8x + 1) \div (x + 1)$?

Explanation:

Step1: Apply Remainder Theorem

For polynomial $f(x)$ divided by $(x-a)$, remainder is $f(a)$. Here, divisor is $(x+1)=(x-(-1))$, so $a=-1$.

Step2: Define the polynomial

Let $f(x)=x^3 - 5x^2 - 8x + 1$

Step3: Substitute $x=-1$ into $f(x)$

$f(-1)=(-1)^3 - 5(-1)^2 - 8(-1) + 1$

Step4: Calculate each term

$(-1)^3=-1$, $-5(-1)^2=-5(1)=-5$, $-8(-1)=8$, so:
$f(-1)=-1 - 5 + 8 + 1$

Step5: Compute final value

$f(-1)=(-1-5)+(8+1)=-6+9=3$

Answer:

3