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

Question
Polynomial factorization
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

Relevant Questions

asked 2021-02-16
Use the factor theorem to determine if the given binomial is a factor of f(x).
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}+{8}{x}^{{3}}+{11}{x}^{{2}}-{11}{x}+{3},{x}+{3}\)
asked 2020-11-09
Complete Factorization Factor the polynomial completely, and find all its zeros.State the multiplicity of each zero.
\(P(x) = x^{4}+2x^{2}+1\)
asked 2020-11-08
Use the linear factorization theorem to construct a polynomial function with the given condition.
n=3 with real coefficients, zeros = -4, 2+3i.
asked 2020-12-24
Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although \(20x^{3}\) appears in both \(20x^{3} + 8x^{2} and 20x^{3} + 10x\), I’ll need to factor \(20x^{3}\) in different ways to obtain each polynomial’s factorization?
asked 2020-11-01
The term \(20x^{3}\) appears in both the expressions \(20x^{3} + 8x^{2}\) and \(20x^{3} + 10x\) but factor \(20x^{3}\) in different ways to obtain each polynomial factorization Whether the given statement make sense or does not make sence.
asked 2021-01-31
Use the prime factorizations \(\displaystyle{a}={2}^{{4}}\times{3}^{{4}}\times{5}^{{2}}\times{7}^{{3}}{\quad\text{and}\quad}{b}={2}^{{2}}\times{3}\times{5}^{{3}}\times{11}\) to find the prime factorization of the following.
(a) LCM(a, b)
(b) GCF(a, b)
asked 2020-10-20
Try to complete Factorization Factor the polynomial completely , and find all its zeros. State the multiplicity of each zero.
\(P(x)=x^{4}-625\)
asked 2020-12-02
Need to calculate:The factorization of \(2x^{3}+6x^{2}+x+3\).
asked 2020-12-24
Need to calculate:The factorization of \(x^{3}+3x^2+2x+6\)
asked 2021-02-21
Solve. The factorization of the polynomial is \((3x+2)(2x^{2}+1)\).
Given Information:
The provided polynomial is \(6x^{3}+4x^{2}+3x+2\).
...