Question

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

Polynomial graphs
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asked 2021-01-22
Graph each polynomial function. Factor first if the expression is not in factored form.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}+{1}\right)}{\left({x}-{1}\right)}\)

Answers (1)

2021-01-23

\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}+{1}\right)}{\left({x}-{1}\right)}\) The function in factored form
The function has three zeros 0, -1, and 1
So, the graph of f(x) crosses the x-axis at (0,0), (-1,0), and (1,0)
To find the y-intercept, substitute 0 for x in f(x)
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}+{1}\right)}{\left({x}-{1}\right)}\)
\(\displaystyle{f{{\left({0}\right)}}}={\left({0}\right)}^{{{2}}}{\left({0}+{1}\right)}{\left({0}-{1}\right)}\) Substitute 0 for x
\(\displaystyle={0}\)
So, the function f(x) crosses the y-axis at (0,0)
\(x^{2} \cdot x \cdot x=x^{4}\)
The leading coefficient is 1
Since the leading coefficient is positive and the function f(x) of degree 4 (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
image

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