Chapter 2.2, Problem 55E

To determine

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

Answer to Problem 55E

The zeros of polynomial function $f\left(t\right)=t^3-4t^2+4t$ are $t=0$ and $t=2$ of multiplicity 1 and 2 respectively. Theresults are verified with the graph.

Explanation of Solution

Given:

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

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 $f\left(t\right)=t^3-4t^2+4t$ may be written as

  $f\left(t\right)=t^3-4t^2+4t$

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

  $\Rightarrow f\left(t\right)=t{\left(t-2\right)}^2$

  $\Rightarrow f\left(t\right)={\left(t-0\right)}^1{\left(t-2\right)}^2$

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

  $\Rightarrow \left(t-0\right){\left(t-2\right)}^2=0$

Now, $\left(t-0\right)=0\Rightarrow t=0,{\left(t-2\right)}^2=0\Rightarrow t=2$

Thus, $t=0$ and $t=2$ are the zeros of function $f\left(t\right)=t^3-4t^2+4t$ of multiplicity 1 and 2 respectively.

Now, we verify the result with graph:

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

  

The graph of function $f\left(t\right)=t^3-4t^2+4t$ intersects t-axis at points $\left(0,0\right)$ and touches at point $\left(2,0\right)$ where two t- intercepts are same. Thus, we have $f\left(0\right)=0$

  $f\left(2\right)=0$ and $f\left(2\right)=0$ .

Hence, $t=0$ and $t=2$ are zeros of polynomial function $f\left(t\right)=t^3-4t^2+4t$ e of multiplicity 1 and 2 respectively.

Conclusion:

The zeros of polynomial function $f\left(t\right)=t^3-4t^2+4t$ are $t=0$ and $t=2$ of multiplicity 1 and 2 respectively. Theresultsare verified with the graph.