Chapter 2.2, Problem 43E

a.

To determine

To find: The zeros of polynomial function algebraically.

Answer to Problem 43E

The zeros of the function $f\left(x\right)=2x^4-2x^2-40$ are $x=\pm \sqrt{5}$ , other zeros are imaginary.

Explanation of Solution

Given:

The polynomial function is $f\left(x\right)=2x^4-2x^2-40$ .

Concept used:

The zeros of a polynomial function $f\left(x\right)$ are the values of x for which $f\left(x\right)=0$ . If monomial $\left(x-a\right)$ , $\left(x-b\right)$ , $\left(x-c\right)$ , … are the factors of function then $x=a$ , $x=b$ …, are the zeros of $f\left(x\right)$ .

Calculations:

The zeros of a polynomial function $f\left(x\right)=2x^4-2x^2-40$ may be written as

  $f\left(x\right)=2\left(x^4-x^2-20\right)$

  $\Rightarrow f\left(x\right)=2\left(x^4-5x^2+4x^2-20\right)$

  $\Rightarrow f\left(x\right)=2\left(x^2-5\right)\left(x^2+4\right)$

The zeros of $f\left(x\right)$ are the values of xfor which $f\left(x\right)=0$

  $\Rightarrow 2\left(x^2-5\right)\left(x^2+4\right)=0$

  $\Rightarrow \left(x^2-5\right)=0\Rightarrow x=\pm \sqrt{5}$ , ${\left(x^2+4\right)}^2=0\Rightarrow x=\pm \sqrt{-4}$ imaginary numbers.

Thus, algebraically calculated zeros of the function $f\left(x\right)=2x^4-2x^2-40$ are $x=\pm \sqrt{5}$ ,and imaginary numbers $x=\pm \sqrt{-4}$

Conclusion:

The zeros of the function $f\left(x\right)=2x^4-2x^2-40$ are $x=\pm \sqrt{5}$ , other zeros are imaginary.