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

Graph each polynomial function. Factor first if the expression is not in factored form.
$f\left(x\right)=2{x}^{3}\left({x}^{2}-4\right)\left(x-1\right)$
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Step 1
$f\left(x\right)=2{x}^{3}\left({x}^{2}-4\right)\left(x-1\right)$
$f\left(x\right)=2{x}^{3}\left(x-2\right)\left(x+2\right)\left(x-1\right)$ 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)
$f\left(x\right)=2{x}^{3}\left({x}^{2}-4\right)\left(x-1\right)$
$f\left(0\right)=2{\left(0\right)}^{3}\left({0}^{2}-4\right)\left(0-1\right)$ Substitute 0 for x
$=0$
So, the function f(x) crosses the y-axis at (0,0
Step 2 $2{x}^{3}\cdot {x}^{2}\cdot x=2{x}^{6}$
Since the leading coefficient is positive and the function f(x) of degree 6 (even degree)
So, the end behavior is
$x\to \mathrm{\infty },f\left(x\right)\to \mathrm{\infty }$
$x\to -\mathrm{\infty },f\left(x\right)\to \mathrm{\infty }$