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# Solving Polynomial Equations

## Solving Polynomial Equations 1.14 - Solution

We want to find the zeros and sketch the graph of the given polynomial function. $\begin{gathered} f(x)=x^6-11x^5+30x^4 \end{gathered}$ Let's do these things one at a time.

### Zeros of the Function

To find the zeros, we need to find the values of $x$ for which $f(x)=0.$ $\begin{gathered} f(x)=0 \quad \Leftrightarrow \quad x^6-11x^5+30x^4=0 \end{gathered}$ Since the function is not written in factored form, we will begin by factoring the equation.
$x^6-11x^5+30x^4=0$
Factor
$x^4\left(x^2-11x+30\right)=0$
$x^4\left(x^2-5x-6x+30\right)=0$
$x^4\left(x(x-5)-6x+30\right)=0$
$x^4\left(x(x-5)-6(x-5)\right)=0$
$x^4(x-5)(x-6)=0$
Now, we can apply the Zero Product Property.
$x^4(x-5)(x-6)=0$
Solve using the Zero Product Property
$\begin{array}{lc}x^4=0 & \text{(I)}\\ x-5=0 & \text{(II)}\\ x-6=0 & \text{(III)}\end{array}$
$\begin{array}{l}x=0 \\ x-5=0 \\ x-6=0 \end{array}$
$\begin{array}{l}x=0 \\ x=5 \\ x-6=0 \end{array}$
$\begin{array}{l}x=0 \\ x=5 \\ x=6 \end{array}$
We found that the zeros of the function are $x=0,$ $x=5,$ and $x=6.$

### Graph

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.

$x$ $x^6-11x^5+30x^4$ $f(x)=x^6-11x^5+30x^4$
${\color{#0000FF}{\text{-} 1}}$ $({\color{#0000FF}{\text{-} 1}})^6-11({\color{#0000FF}{\text{-} 1}})^5+30({\color{#0000FF}{\text{-} 1}})^4$ ${\color{#009600}{42}}$
${\color{#0000FF}{2}}$ ${\color{#0000FF}{2}}^6-11({\color{#0000FF}{2}})^5+30({\color{#0000FF}{2}})^4$ ${\color{#009600}{192}}$
${\color{#0000FF}{4}}$ ${\color{#0000FF}{4}}^6-11({\color{#0000FF}{4}})^5+30({\color{#0000FF}{4}})^4$ ${\color{#009600}{512}}$
${\color{#0000FF}{5.5}}$ ${\color{#0000FF}{5.5}}^6-11({\color{#0000FF}{5.5}})^5+30({\color{#0000FF}{5.5}})^4$ $\approx {\color{#009600}{\text{-} 228.8}}$

The points $({\color{#0000FF}{\text{-} 1}},{\color{#009600}{42}}),$ $({\color{#0000FF}{2}},{\color{#009600}{192}}),$ $({\color{#0000FF}{4}},{\color{#009600}{512}}),$ and $({\color{#0000FF}{5.5}},{\color{#009600}{\text{-} 228.8}})$ are on the graph of the function. Now, we will determine the leading coefficient and degree of the polynomial function. $\begin{gathered} f(x)=x^6-11x^5+30x^4 \\ \Updownarrow \\ f(x)=\textcolor{darkorange}{1}x^\textcolor{magenta}{6}-11x^5+30x^4 \end{gathered}$ We can see now that the leading coefficient is $\textcolor{darkorange}{1},$ which is a positive number. Also, the degree is $\textcolor{magenta}{6},$ which is an even number. Therefore, the end behavior is up and up. With this in mind, we will plot the zeros, obtained points, and connect them with a smooth curve.