Chapter 2.4, Problem 55E

Solution

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

  $-\frac{1}{2},4,i,-i$

Given information:

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

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 $x=-\frac{1}{2}$ ,

  $\begin{array}{cccccc}-0.5& 2& -7& -2& -7& -4\\ & \downarrow & -1& 4& -1& 4\\ & 2& -8& 2& -8& \boxed{0}\end{array}$

Since the last number of the bottom rows is 0, the $x=-\frac{1}{2}$ is a real root of the function.

The quotient of the polynomial is $2x^3-8x^2+2x-8$ .

Setup the synthetic division for $x=4$ ,

  $\begin{array}{ccccc}4& 2& -8& 2& -8\\ & \downarrow & 8& 0& 8\\ & 2& 0& 2& \boxed{0}\end{array}$

Since the last number of the bottom rows is 0, the $x=4$ is a real root of the function.

The quotient of the polynomial is $2x^2+2$ .

The other two zeros are,

  $\begin{array}{c}2x^2+2=0\\ x^2=-1\\ x=\pm i\end{array}$