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):

\(\displaystyle{P}{\left({x}\right)}={\left({3}{x}-{1}\right)}{\left({x}^{{{2}}}+{36}\right)}{\left({x}+{3}\right)}{\left({x}-{3}\right)}\)

\(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{x}^{{{4}}}+{81}{x}^{{{3}}}-{27}{x}^{{{2}}}-{972}{x}+{324}\)

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):

\(\displaystyle{P}{\left({x}\right)}={\left({3}{x}-{1}\right)}{\left({x}^{{{2}}}+{36}\right)}{\left({x}+{3}\right)}{\left({x}-{3}\right)}\)