Question

# Use the factorization theorem to determine whether x−1/2 is a factor of f(x) = 2x^4 − x^3 + 2x − 1.

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

2021-02-25
Step 1
If $$\displaystyle{x}-\frac{{1}}{{2}}$$ is a factor then the remainder when f(x) divided by it will be zero.
Put $$\displaystyle{x}-\frac{{1}}{{2}}={0}$$
$$\displaystyle{x}=\frac{{1}}{{2}}$$
Step 2
Substitute $$\displaystyle{x}=\frac{{1}}{{2}}\in{f{{\left({x}\right)}}}$$.
$$\displaystyle{f{{\left(\frac{{1}}{{2}}\right)}}}={2}{\left(\frac{{1}}{{2}}\right)}^{{4}}-{\left(\frac{{1}}{{2}}\right)}^{{3}}+{2}{\left(\frac{{1}}{{2}}\right)}-{1}$$
$$\displaystyle=\frac{{1}}{{8}}-\frac{{1}}{{8}}+{1}-{1}$$
=0+0
=0
Thus the remainder is zero.
$$\displaystyle{x}-\frac{{1}}{{2}}$$ is a factor of f(x).