Question

Using remainder theorem, find the remainder on dividing $f\left(x\right)$ by $(x+3)$, where:
$f\left(x\right)=3x^3+7x^2-5x+1$.

Answer 1

We know, the remainder theorem states that if a polynomial $f\left(x\right)$ is divided by $(x-a)$, the remainder is $f\left(a\right)$.

Here, we have the polynomial $f\left(x\right)=3x^3+7x^2-5x+1$ which is divided by $(x+3)$, therefore, by remainder theorem, the remainder $R$ is:

$R=f(-3)=3(-3{)}^3+7(-3{)}^2-5(-3)+1=3\times (-27)+(7\times 9)+15+1=-81+63+15+1=79-81=-2.$

Hence, the remainder is $-2$.