What is the remainder when $P(x) = -x^3 - 2x^2 - 3x + 8$ is divided by $x + 2$? - HSC - SSCE Mathematics Extension 1 - Question 3 - 2021 - Paper 1

To find the remainder when dividing by $x + 2$, we can use the Remainder Theorem. According to the theorem, the remainder of the polynomial $P(x)$ when divided by $x - a$ is simply $P(a)$. Here, we want to find $P(-2)$:

P(-2) &= -(-2)^3 - 2(-2)^2 - 3(-2) + 8 \\ &= -(-8) - 2(4) + 6 + 8 \\ &= 8 - 8 + 6 + 8 \\ &= 14. \end{align*}$$ Thus, the remainder is 14.