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{{ printedBook.courseTrack.name }} {{ printedBook.name }} We want to find the zeros and sketch the graph of the given polynomial function. $f(x)=-2x_{5}+2x_{4}+40x_{3} $ Let's do these things one at a time.

$-2x_{5}+2x_{4}+40x_{3}=0$

$-2x_{3}(x+4)(x−5)=0$

$-2x_{3}(x+4)(x−5)=0$

Solve using the Zero Product Property

ZeroProdPropUse the Zero Product Property

$-2x_{3}=0x+4=0x−5=0 $

$x_{3}=0x+4=0x−5=0 $

$x=0x+4=0x−5=0 $

$x=0x=-4x−5=0 $

$x=0x=-4x=5 $

To draw the graph of the function, we will find some additional points and consider the end behavior. Let's use a table to find additional points.

$x$ | $-2x_{5}+2x_{4}+40x_{3}$ | $f(x)=-2x_{5}+2x_{4}+40x_{3}$ |
---|---|---|

$-5$ | $-2(-5)_{5}+2(-5)_{4}+40(-5)_{3}$ | $2500$ |

$-2$ | $-2(-2)_{5}+2(-2)_{4}+40(-2)_{3}$ | $-224$ |

$3$ | $-2(3)_{5}+2(3)_{4}+40(3)_{3}$ | $756$ |

$5.5$ | $-2(5.5)_{5}+2(5.5)_{4}+40(5.5)_{3}$ | $≈-1581$ |

The points $(-5,2500),$ $(-2,-224),$ $(3,756),$ and $(5.5,-1581)$ are on the graph of the function. Now, we will determine the leading coefficient and degree of the polynomial function.
$f(x)=-2x_{5}+2x_{4}+40x_{3} $
We can see now that the leading coefficient is $-2,$ which is a negative number. Also, the degree is $5,$ which is an odd number. Therefore, the end behavior is **up** and **down**. With this in mind, we will plot the zeros, the obtained points, and graph the function.