Question

Find the remainder when $x^3+3x^2+3x+1$ is divided by $x+\pi$.

Answer 1

Evaluate the remainder:

Given that the polynomial $x^3+3x^2+3x+1$ is divided by $x+\pi$.

Let $P\left(x\right)=x^3+3x^2+3x+1$

According to the remainder theorem, when a polynomial, $P\left(x\right)$ is divided by a linear polynomial, $x-a$, the remainder of that division will be equivalent to $P\left(a\right)$.

Thus, if $x^3+3x^2+3x+1$ is divided by the linear polynomial $x+\pi$, the remainder of that division will be equivalent to $P(-\pi )$.

Hence, the remainder

$P(-\pi )={(-\pi )}^3+3{(-\pi )}^2+3(-\pi )+1$

$=-{\pi }^3+3{\pi }^2-3\pi +1$

Therefore, when $x^3+3x^2+3x+1$ is divided by $x+\pi$, we get the remainder $-{\pi }^3+3{\pi }^2-3\pi +1$.