# Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=2x^{3}(x^{2}−4)(x−1)

Question
Polynomial graphs
Graph each polynomial function. Factor first if the expression is not in factored form.
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}−{4}\right)}{\left({x}−{1}\right)}$$

2020-11-21
Step 1
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}-{4}\right)}{\left({x}-{1}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}-{2}\right)}{\left({x}+{2}\right)}{\left({x}-{1}\right)}$$ The function in factored form
The function has four zeros 0,2,-2 and 1
So, the graph of f(x) crosses the x-axis at (0,0), (2,0), (-2,0) and (1,0)
To find the y-intercept, subtitute 0 for x in f(x)
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}^{{{2}}}-{4}\right)}{\left({x}-{1}\right)}$$
$$\displaystyle{f{{\left({0}\right)}}}={2}{\left({0}\right)}^{{{3}}}{\left({0}^{{{2}}}-{4}\right)}{\left({0}-{1}\right)}$$ Substitute 0 for x
$$\displaystyle={0}$$
So, the function f(x) crosses the y-axis at (0,0
Step 2 $$\displaystyle{2}{x}^{{{3}}}\cdot{x}^{{{2}}}\cdot{x}={2}{x}^{{{6}}}$$
Since the leading coefficient is positive and the function f(x) of degree 6 (even degree)
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 below
$$\displaystyle{h}{\mathtt{{p}}}{s}:{/}{q}{2}{a}.{s}{3}-{u}{s}-{w}{e}{s}{t}-{1}.{a}{m}{a}{z}{o}{n}{a}{w}{s}.{c}{o}\frac{{m}}{{d}}{e}\frac{{v}}{{19610300981}}.{j}{p}{g}$$

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