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
ANSWERED
asked 2021-02-24
Use the factorization theorem to determine whether \(\displaystyle{x}−\frac{{1}}{{2}}\) is a factor
of \(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{4}}−{x}^{{3}}+{2}{x}−{1}\).

Answers (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).
0
 
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