Chapter 2.2, Problem 56E

To determine

To find: The zeros and their multiplicities of given polynomial function, and verify the results with graphing utility.

Answer to Problem 56E

The zeros of polynomial function $f\left(x\right)=x^4-x^3-20x^2$ are $x=0$ of multiplicity 2 and $x=-4$ , $x=5$ each of multiplicity 1. Theresults are verified with the graph.

Explanation of Solution

Given:

The polynomial function is $f\left(x\right)=x^4-x^3-20x^2$ .

Concept used:

The zeros of a polynomial function $f\left(x\right)$ are the values of x for which $f\left(x\right)=0$ . If monomial $\left(x-a\right),\left(x-b\right),\left(x-c\right)\dots$ , are the factors of function then $x=a,x=b,\dots$ , are the zeros of $f\left(x\right)$ . For a polynomial function, a factor of ${\left(x-a\right)}^m$

  $m>0$ , yields a repeated zero

  $x=a$ , of multiplicity m.

Calculations:

The zeros of a polynomial function $f\left(x\right)=x^4-x^3-20x^2$ may be written as

  $f\left(x\right)=x^2\left(x^2-x-20\right)$

  $\Rightarrow f\left(x\right)=x^2\left(x^2-5x+4x-20\right)$

  $\Rightarrow f\left(x\right)=x^2\left(x\left(x-5\right)+4\left(x-5\right)\right)$

  $\Rightarrow f\left(x\right)={\left(x-0\right)}^2\left(x-5\right)\left(x+4\right)$

The zeros of function are values of x for which $f\left(x\right)=0$

  $\Rightarrow {\left(x-0\right)}^2\left(x-5\right)\left(x+4\right)=0$

Now, $\left(x-0\right)=0\Rightarrow x=0,\left(x+4\right)=0\Rightarrow x=-4$ , and $\left(x-5\right)=0\Rightarrow x=5$

Thus, $x=0$ is the zero of function $f\left(x\right)$ of multiplicity 2 and $x=-4$ , $x=5$ are zeroseach of multiplicity 1.

Now, we verify the result with graph:

The graph of function $f\left(x\right)=x^4-x^3-20x^2$ is shown in Figure-1 here.

  

The graph of function $f\left(x\right)=x^4-x^3-20x^2$ intersects x-axis at points $\left(0,0\right)$

  $\left(-4,0\right)$ , and $\left(5,0\right)$ . Thus, we have $f\left(0\right)=0$

  $f\left(-4\right)=0$ and $f\left(5\right)=0$ .

Hence, $x=0$ is the zero of function $f\left(x\right)$ of multiplicity 2 and $x=-4$ , $x=5$ are zeros each of multiplicity 1.

Conclusion:

The zeros of polynomial function $f\left(x\right)=x^4-x^3-20x^2$ are $x=0$ of multiplicity 2 and $x=-4$ , $x=5$ each of multiplicity 1. Theresultsare verified with the graph.