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).

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).