Chapter 2.4, Problem 54E

Solution

To find: All the zeros of the function $f\left(x\right)=x^4-x^3-7x^2+5x+10$ .

  $-1,2,\sqrt{5},-\sqrt{5}$

Given information:

  $f\left(x\right)=x^4-x^3-7x^2+5x+10$

Concept Used:

Divide a polynomial function $g\left(x\right)$ by $\left(x-k\right)$ , using synthetic division and the number at last is zero, then $k$ is a real root of the equation $g\left(x\right)=0$ .

Calculation:

Setup the synthetic division for $x=-1$ ,

  $\begin{array}{cccccc}-1& 1& -1& -7& 5& 10\\ & \downarrow & -1& 2& 5& -10\\ & 1& -2& -5& 10& \boxed{0}\end{array}$

Since the last number of the bottom rows is 0, the $x=-1$ is a real root of the function.

The quotient of the polynomial is $x^3-2x^2-5x+10$ .

Setup the synthetic division for $x=2$ ,

  $\begin{array}{ccccc}2& 1& -2& -5& 10\\ & \downarrow & 2& 0& -10\\ & 1& 0& -5& \boxed{0}\end{array}$

Since the last number of the bottom rows is 0, the $x=2$ is a real root of the function.

The quotient of the polynomial is $x^2-5$ .

The other two zeros are,

  $\begin{array}{c}{x}^2-5=0\\ x^2=5\\ x=\pm \sqrt{5}\end{array}$