Chapter 4.6, Problem 10E

To determine

To find: all zeros of the polynomial function.

Answer to Problem 10E

The zeros are $1,2,-3$ and $-5$ .

Explanation of Solution

Given information:

Given function,

  $f\left(x\right)=x^4+5x^3-7x^2-29x+30$

Calculation:

Consider the equation $f\left(x\right)=x^4+5x^3-7x^2-29x+30$ :

Because $f\left(x\right)$ is a polynomial function of degree 4, so it has 4 zeros.

The possible rational zeros are $\pm 1,\pm 2,\pm 3,\pm 5,\pm 6,\pm 10,\pm 15\text{and}\pm 30.$

Choose reasonable values from the list above to test using the graph of the function.

  

From the above graph the values are $x=-5,x=-3,x=1\text{and}x=2$ .

Test out the polynomial using synthetic division until a zero is found.

  $\begin{array}{cccccc}1& 1& 5& -7& -29& 30\\ & & 1& 6& -1& -30\\ & 1& 6& -1& -30& 0\end{array}$

Factor out the polynomial using the result of above synthetic division.

  $\begin{array}{c}f\left(x\right)=x^4+5x^3-7x^2-29x+30\\ =\left(x-1\right)\left(x^3+6x^2-x-30\right)\end{array}$

Repeat the steps above for $g\left(x\right)=x^3+6x^2-x-30$ .

The possible rational are $\pm 1,\pm 2,\pm 3,\pm 5,\pm 6,\pm 10,\pm 15\text{and}\pm 30.$ Test the value using synthetic division until a zero is found.

  $\begin{array}{ccccc}2& 1& 6& -1& -30\\ & & 2& 16& 30\\ & 1& 8& 15& 0\end{array}$

Factor out the polynomial using the result of above synthetic division.

  $\begin{array}{c}f\left(x\right)=\left(x-1\right)\left(x^3+6x^2-x-30\right)\\ =\left(x-1\right)\left(x-2\right)\left(x^2+8x+15\right)\\ =\left(x-1\right)\left(x-2\right)\left(x+3\right)\left(x+5\right)\end{array}$

Solve $f\left(x\right)=0$ .

  $\begin{array}{c}\left(x-1\right)\left(x-2\right)\left(x+3\right)\left(x+5\right)=0\\ x=1,2,-3,-5\end{array}$

Therefore, the zeros are $1,2,-3$ and $-5$ .