# Use the factorization theorem to determine whether x-\frac{1}{2} is a factor of f(x) = 2x^{4}-x^{3}=2x-1.

Polynomial factorization
Use the factorization theorem to determine whether $$x-\frac{1}{2}$$ is a factor of $$f(x) = 2x^{4}-x^{3}=2x-1$$.

2021-05-31

Step 1
If $$x-\frac{1}{2}$$ is a factor the the remainder when f(x) divided by it will be zero.
Put $$x-\frac{1}{2}=0$$
$$x=\frac{1}{2}$$
Step 2
Substitute $$x=\frac{1}{2}$$ in f(x).
$$f(\frac{1}{2})=2(\frac{1}{2})^{4}-(\frac{1}{2})^{3}+2(\frac{1}{2})-1$$
$$=\frac{1}{8}-\frac{1}{8}+1-1$$
$$=0+0$$
$$=0$$
Thus the remainder is zero.
$$x-\frac{1}{2}$$ is a factor of f(x)