Chapter 3, Problem 20CLT

To determine

Find the zeros of the function and write the function as a product of linear factors.

Answer to Problem 20CLT

The linear factors are $x^2,\left(x+7\right)$ and $\left(x-3\right)$ . The zeros for the function are $-7,0$ and $3$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=x^4+4x^3-21x^2$ .

Calculation:

Linear factors of the function $f\left(x\right)=x^4+4x^3-21x^2$ is

  $\begin{array}{l}f\left(x\right)=x^4+4x^3-21x^2\\ =x^2\left(x^2+4x-21\right)\\ =x^2\left(x^2+7x-3x-21\right)\\ =x^2\left\{x\left(x+7\right)-3\left(x+7\right)\right\}\\ =x^2\left(x+7\right)\left(x-3\right)\end{array}$

For zeros

  $x^2\left(x+7\right)\left(x-3\right)=0$

Use zero factor principal.

  $\begin{array}{l}{x}^2=0\\ \Rightarrow x=0\end{array}$

  $\begin{array}{l}\left(x+7\right)=0\\ \Rightarrow x+7-7=0-7\\ \Rightarrow x=-7\\ \text{and}\\ x-3=0\\ \Rightarrow x-3+3=0+3\\ \Rightarrow x=3\end{array}$

Hence the linear factors are $x^2,\left(x+7\right)$ and $\left(x-3\right)$ . The zeros for the function are $-7,0$ and $3$ .