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Solving Polynomial Equations

Solving Polynomial Equations 1.14 - Solution

arrow_back Return to Solving Polynomial Equations

We want to find the zeros and sketch the graph of the given polynomial function. f(x)=x611x5+30x4\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 xx for which f(x)=0.f(x)=0. f(x)=0x611x5+30x4=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.
x611x5+30x4=0x^6-11x^5+30x^4=0
Factor
x4(x211x+30)=0x^4\left(x^2-11x+30\right)=0
x4(x25x6x+30)=0x^4\left(x^2-5x-6x+30\right)=0
x4(x(x5)6x+30)=0x^4\left(x(x-5)-6x+30\right)=0
x4(x(x5)6(x5))=0x^4\left(x(x-5)-6(x-5)\right)=0
x4(x5)(x6)=0x^4(x-5)(x-6)=0
Now, we can apply the Zero Product Property.
x4(x5)(x6)=0x^4(x-5)(x-6)=0
Solve using the Zero Product Property
x4=0(I)x5=0(II)x6=0(III)\begin{array}{lc}x^4=0 & \text{(I)}\\ x-5=0 & \text{(II)}\\ x-6=0 & \text{(III)}\end{array}
x=0x5=0x6=0\begin{array}{l}x=0 \\ x-5=0 \\ x-6=0 \end{array}
x=0x=5x6=0\begin{array}{l}x=0 \\ x=5 \\ x-6=0 \end{array}
x=0x=5x=6\begin{array}{l}x=0 \\ x=5 \\ x=6 \end{array}
We found that the zeros of the function are x=0,x=0, x=5,x=5, and x=6.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.

xx x611x5+30x4x^6-11x^5+30x^4 f(x)=x611x5+30x4f(x)=x^6-11x^5+30x^4
-1{\color{#0000FF}{\text{-} 1}} (-1)611(-1)5+30(-1)4({\color{#0000FF}{\text{-} 1}})^6-11({\color{#0000FF}{\text{-} 1}})^5+30({\color{#0000FF}{\text{-} 1}})^4 42{\color{#009600}{42}}
2{\color{#0000FF}{2}} 2611(2)5+30(2)4{\color{#0000FF}{2}}^6-11({\color{#0000FF}{2}})^5+30({\color{#0000FF}{2}})^4 192{\color{#009600}{192}}
4{\color{#0000FF}{4}} 4611(4)5+30(4)4{\color{#0000FF}{4}}^6-11({\color{#0000FF}{4}})^5+30({\color{#0000FF}{4}})^4 512{\color{#009600}{512}}
5.5{\color{#0000FF}{5.5}} 5.5611(5.5)5+30(5.5)4{\color{#0000FF}{5.5}}^6-11({\color{#0000FF}{5.5}})^5+30({\color{#0000FF}{5.5}})^4 -228.8\approx {\color{#009600}{\text{-} 228.8}}

The points (-1,42),({\color{#0000FF}{\text{-} 1}},{\color{#009600}{42}}), (2,192),({\color{#0000FF}{2}},{\color{#009600}{192}}), (4,512),({\color{#0000FF}{4}},{\color{#009600}{512}}), and (5.5,-228.8)({\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. f(x)=x611x5+30x4f(x)=1x611x5+30x4\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 1,\textcolor{darkorange}{1}, which is a positive number. Also, the degree is 6,\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.