 Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=2x^{3}+7x^{2}+4x-4 Ayaana Buck 2021-03-03 Answered
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}$$

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Step 1
we have to find all the rational zeros of the polynomial and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}$$
Step 2
$$\displaystyle{2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}$$
$$\displaystyle{\left({2}{x}-{1}\right)}{\left({x}^{{{2}}}+{4}{x}+{4}\right)}$$
$$\displaystyle{\left({2}{x}-{1}\right)}{\left({x}+{2}\right)}^{{{2}}}{\left({2}{x}-{1}\right)}{\left({x}+{2}\right)}{\left({x}+{2}\right)}$$
therefore
$$\displaystyle{2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}={\left({2}{x}-{1}\right)}{\left({x}+{2}\right)}{\left({x}+{2}\right)}$$
now as we have to find the rational roots
therefore
$$(2x-1)(x+2)(x+2)=0$$
$$\displaystyle\Rightarrow{x}=-{2},{x}=-{2},{x}={\frac{{{1}}}{{{2}}}}$$