To find: the real zeros of polynomial function form an arithmetic sequaence f(x)=x^{4}-4x^{3}-4x^{2}+16x

FobelloE 2021-08-07 Answered
To find:
The real zeros of polynomial function form an arithmetic sequaence
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}-{4}{x}^{{{3}}}-{4}{x}^{{{2}}}+{16}{x}\)

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Expert Answer

davonliefI
Answered 2021-08-08 Author has 23740 answers
A ration expression is a fraction that is quotient of two polynomials. A rational function is defined by the two polynomial functions.
A function f of the form \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}}\) is a rational function.
Where, \(\displaystyle{p}{\left({x}\right)}\) and \(\displaystyle{q}{\left({x}\right)}\) are polynomial functions, with \(\displaystyle{q}{\left({x}\right)}\ne{q}{0}.\)
Calculation:
The given polinomial function form an arithmetic sequence is
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}-{4}{x}^{{{3}}}-{4}{x}^{{{2}}}+{16}{x}\)
Here, the constant is 0.
The above equatioon can be rewritten as
\(\displaystyle{f{{\left({x}\right)}}}={x}{\left({x}^{{{3}}}-{4}{x}^{{{2}}}-{4}{x}+{16}\right)}\)
The possiblities for \(\displaystyle{\frac{{{p}}}{{{q}}}}\) are \(\displaystyle\pm{1},\ \pm{2},\ \pm{4}\) and \(\displaystyle\pm{8}\)
Factoring the term \(\displaystyle{\left({x}^{{{3}}}-{4}{x}^{{{2}}}-{4}{x}+{16}\right)},\) we get
\(\displaystyle{\left({x}^{{{3}}}-{4}{x}^{{{2}}}-{4}{x}+{16}\right)}={\left({x}+{2}\right)}{\left({x}^{{{2}}}-{6}{x}+{8}\right)}\)
Factoring the term \(\displaystyle{\left({x}^{{{2}}}-{6}{x}+{8}\right)},\) we get
\(\displaystyle{\left({x}^{{{2}}}-{6}{x}+{8}\right)}={\left({x}-{2}\right)}{\left({x}-{4}\right)}\)
Combining all the terms, we get
\(\displaystyle{\left({x}^{{{3}}}-{4}{x}^{{{2}}}-{4}{x}+{16}\right)}={\left({x}+{2}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}\)
So, \(\displaystyle{f{{\left({x}\right)}}}={x}{\left({x}^{{{3}}}-{4}{x}^{{{2}}}-{4}{x}+{16}\right)}={x}{\left({x}+{2}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}\)
Thus, the real zeros are \(\displaystyle-{2},\ {0},\ {2},\) and \(\displaystyle{4}.\)
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