# Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=2x^{3}-3x^{2}-2x+3

Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}$$

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Arham Warner
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 $$\displaystyle{\frac{{{3}}}{{{2}}}}$$.