Step 1

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

All rational zeros = ?

Factored form = ?

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

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

\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}\)

The polynomial in factored form:

\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}\)

Step 2

To find rational zeros:

Put P(x) = 0

\(\displaystyle{\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}={0}\)

\(\displaystyle{2}{x}-{3}={0}\Rightarrow{x}={\frac{{{3}}}{{{2}}}}\)

\(\displaystyle{x}-\sqrt{{{2}}}={0}\Rightarrow{x}=\sqrt{{{2}}}\)

\(\displaystyle{x}+\sqrt{{{2}}}={0}\Rightarrow{x}=-\sqrt{{{2}}}\)

The rational zeros are: \(\displaystyle-\sqrt{{{2}}},\sqrt{{{2}}}\) and \(\displaystyle{\frac{{{3}}}{{{2}}}}\).

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

All rational zeros = ?

Factored form = ?

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

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

\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}\)

The polynomial in factored form:

\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}\)

Step 2

To find rational zeros:

Put P(x) = 0

\(\displaystyle{\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}={0}\)

\(\displaystyle{2}{x}-{3}={0}\Rightarrow{x}={\frac{{{3}}}{{{2}}}}\)

\(\displaystyle{x}-\sqrt{{{2}}}={0}\Rightarrow{x}=\sqrt{{{2}}}\)

\(\displaystyle{x}+\sqrt{{{2}}}={0}\Rightarrow{x}=-\sqrt{{{2}}}\)

The rational zeros are: \(\displaystyle-\sqrt{{{2}}},\sqrt{{{2}}}\) and \(\displaystyle{\frac{{{3}}}{{{2}}}}\).