Question

Given P(x)=3x^{5}−5x^{4}+37x^{3}−83x^{2}−176x−48, and that 4i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including P(x)=.

Factors and multiples
ANSWERED
asked 2021-08-11
Given \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}−{5}{x}^{{{4}}}+{37}{x}^{{{3}}}−{83}{x}^{{{2}}}−{176}{x}−{48}\), and that 4i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including \(\displaystyle{P}{\left({x}\right)}=\).

Answers (1)

2021-08-12
Step 1
The given polynomial is:
\(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}−{5}{x}^{{{4}}}+{37}{x}^{{{3}}}−{83}{x}^{{{2}}}−{176}{x}−{48}\)
Step 2
Now \(\displaystyle{x}={4}{i}\) is a zero of the given polynomial, therefore \(\displaystyle{x}=-{4}{i}\) is also zero of the given polynomial.
Therefore: \(\displaystyle{\left({x}-{4}{i}\right)}{\left({x}+{4}{i}\right)}\) divides the given polynomial. That is one of the factors of given polynomial is: \(\displaystyle{x}{2}+{16}\)
\(\displaystyle{\frac{{{P}{\left({x}\right)}}}{{{x}^{{{2}}}+{16}}}}={\frac{{{3}{x}^{{{5}}}-{5}{x}^{{{4}}}+{37}{x}^{{{3}}}-{83}{x}^{{{2}}}-{176}{x}-{48}}}{{{x}^{{{2}}}+{16}}}}\)
\(\displaystyle{\frac{{{P}{\left({x}\right)}}}{{{x}^{{{2}}}+{16}}}}={3}{x}^{{{3}}}-{5}{x}^{{{2}}}-{11}{x}-{3}\)
\(\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{16}\right)}{\left({3}{x}^{{{3}}}-{5}{x}^{{{2}}}-{11}{x}-{3}\right)}\)
\(\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{16}\right)}{\left({x}+{1}\right)}{\left({x}-{3}\right)}{\left({3}{x}+{1}\right)}\)
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