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
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 $$\displaystyle{P}{\left({x}\right)}=$$.

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: $$\displaystyle{x}{2}+{16}$$
$$\displaystyle{\frac{{{P}{\left({x}\right)}}}{{{x}^{{{2}}}+{16}}}}={\frac{{{3}{x}^{{{5}}}-{5}{x}^{{{4}}}+{37}{x}^{{{3}}}-{83}{x}^{{{2}}}-{176}{x}-{48}}}{{{x}^{{{2}}}+{16}}}}$$
$$\displaystyle{\frac{{{P}{\left({x}\right)}}}{{{x}^{{{2}}}+{16}}}}={3}{x}^{{{3}}}-{5}{x}^{{{2}}}-{11}{x}-{3}$$
$$\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{16}\right)}{\left({3}{x}^{{{3}}}-{5}{x}^{{{2}}}-{11}{x}-{3}\right)}$$
$$\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{16}\right)}{\left({x}+{1}\right)}{\left({x}-{3}\right)}{\left({3}{x}+{1}\right)}$$