Chapter 4.4, Problem 22E

To determine

To find: The number of possible positive real zeros and the number of possible negative real zeros for function $f\left(x\right)=x^4-5x^2+4$ . Then determine the rational zeros.

Answer to Problem 22E

Possible zeros: $\pm 1\text{, }\pm 2,\pm \text{4}$

  $\text{Zeros: }\pm \text{1,}\pm \text{2}$

Explanation of Solution

Given information: $f\left(x\right)=x^4-5x^2+4$

Calculation:

  $x^4-5x^2+4$

Possible zeros: $\pm 1\text{, }\pm 2,\pm \text{4}$

Descartes Rule of Signs gives:

0 or 2 positive roots, 0 or 2 negative roots.

Using synthetic division we find:

  

So $x=1$ is a zero and:

  $\left(x-1\right)\left(x^3+x^2-4x-4\right)$

Using synthetic division again we find:

  

So $x=-1$ is a zero and

  $\left(x-1\right)\left(x+1\right)\left(x^2-4\right)$ are factors

Finally this gives us:

  $\left(x-1\right)\left(x+1\right)\left(x-2\right)\left(x+2\right)$

So the zeros are: $\pm 1,\pm 2$