Question

# Given P(x)=3x^{5}-x^{4}+81x^{3}-27x^{2}-972x+324, and that 6i 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 $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{x}^{{{4}}}+{81}{x}^{{{3}}}-{27}{x}^{{{2}}}-{972}{x}+{324}$$, and that 6i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including $$\displaystyle{P}{\left({x}\right)}=$$.

2021-08-03
Step 1
$$\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{x}^{{{4}}}+{81}{x}^{{{3}}}-{27}{x}^{{{2}}}-{972}{x}+{324}$$
Given: 6i is a zero.
$$\displaystyle\Rightarrow-{6}{i}$$ is also a zero.
$$\displaystyle\Rightarrow{\left({x}+{6}{i}\right)}{\left({x}-{6}{i}\right)}$$ in a factor
$$\displaystyle\Rightarrow{\left({x}^{{{2}}}+{36}\right)}$$ in a factor of P(x).
Step 2
$$\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{x}^{{{4}}}+{81}{x}^{{{3}}}-{27}{x}^{{{2}}}-{972}{x}+{324}$$
$$\displaystyle={x}^{{{4}}}{\left({3}{x}-{1}\right)}+{27}{x}^{{{2}}}{\left({3}{x}-{1}\right)}-{324}{\left({3}{x}-{1}\right)}$$
$$\displaystyle={\left({3}{x}-{1}\right)}{\left({x}^{{{4}}}+{27}{x}^{{{2}}}-{324}\right)}$$
$$\displaystyle={\left({3}{x}-{1}\right)}{\left[{x}^{{{2}}}{\left({x}^{{{2}}}+{36}\right)}-{9}{\left({x}^{{{2}}}+{36}\right)}\right]}$$
$$\displaystyle={\left({3}{x}-{1}\right)}{\left({x}^{{{2}}}+{36}\right)}{\left({x}^{{{2}}}-{9}\right)}$$
$$\displaystyle={\left({3}{x}-{1}\right)}{\left({x}^{{{2}}}+{36}\right)}{\left({x}+{3}\right)}{\left({x}-{3}\right)}$$
Factored form of P(x):
$$\displaystyle{P}{\left({x}\right)}={\left({3}{x}-{1}\right)}{\left({x}^{{{2}}}+{36}\right)}{\left({x}+{3}\right)}{\left({x}-{3}\right)}$$