Question # For each polynomial function, one zero is given. Find all rational zeros and factor the polynomial. Then graph the function. f(x)=3x^{3}+x^{2}-10x-8, zero:2

Polynomial graphs
ANSWERED For each polynomial function, one zero is given. Find all rational zeros and factor the polynomial. Then graph the function. $$f(x)=3x^{3}+x^{2}-10x-8$$, zero:2 Step 1 Since 2 is a zero of $$\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{{3}}}+{x}^{{{2}}}-{10}{x}-{8}$$ The quotient is $$\displaystyle{3}{x}^{{{2}}}+{7}{x}+{4}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}-{2}\right)}{\left({3}{x}^{{{2}}}+{7}{x}+{4}\right)}$$ Factor the trinomial $$\displaystyle{3}{x}^{{{2}}}+{7}{x}+{4}={\left({3}{x}+{4}\right)}{\left({x}+{1}\right)}$$ So, the function in factored form is $$\displaystyle{f{{\left({x}\right)}}}={\left({x}-{2}\right)}{\left({3}{x}+{4}\right)}{\left({x}+{1}\right)}$$ Set each factor equal to zero $$\displaystyle{x}-{2}={0}$$ or $$\displaystyle{3}{x}+{4}={0}$$ or $$\displaystyle{x}+{1}={0}$$ So, all rational zeros are $$\displaystyle{2},-{\frac{{{4}}}{{{2}}}},-{1}$$ Step 2 Since, all rational zeros are $$\displaystyle{2},-{\frac{{{4}}}{{{3}}}},-{1}$$ So, the graph of the function crosses the x-axis at $$\displaystyle{\left({2},{0}\right)},{\left(-{\frac{{{4}}}{{{3}}}},{0}\right)},{\left(-{1},{0}\right)}$$ To find the y-intercept evaluate f(0) $$\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{{3}}}+{x}^{{{2}}}-{10}{x}-{8}$$
$$\displaystyle{f{{\left({0}\right)}}}={3}{\left({0}\right)}^{{{3}}}+{\left({0}\right)}^{{{2}}}-{10}{\left({0}\right)}-{8}$$ Substitute 0 for x $$\displaystyle=-{8}$$ The leading coefficient is 3 (positive), and the function of degree 3 (odd) So, the end behavior is $$\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty$$ $$\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow-\infty$$ See the graph of f(x) 