# Use the factor theorem to determine if the given binomial is a factor of f(x). f(x)=x^4+8x^3+11x^2-11x+3, x+3

Use the factor theorem to determine if the given binomial is a factor of f(x).
$f\left(x\right)={x}^{4}+8{x}^{3}+11{x}^{2}-11x+3,x+3$
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Step 1
To decide if the given binomial divides the polynomial f(x), using the factor theorem
Step 2
Recall the factor theorem: $\left(x-a\right)$ is a factor of the polynomial f(x) if and only if $f\left(a\right)=0$. Apply this in the present case with $a=-3$, that is , test whether f(-3) is zero or otherwise.
$f\left(x\right)={x}^{4}+8{x}^{3}+11{x}^{2}-11x+3$.
$f\left(-3\right)=81-216+99+33+3=0$
So, $\left(x+3\right)$ is a factor of f(x)
Step 3
Result: $x+3$ is a factor of f(x) (see factorization above)
$f\left(x\right)={x}^{4}+8{x}^{3}+11{x}^{2}-11x+3$
$=\left(x+3\right)\left({x}^{3}+5{x}^{2}-4x+1\right)$