Chapter 2.5, Problem 30E

To determine

Find the zeros of the function $f(x)=x^4+29x^2+100$ and write the polynomial as a product of linear factors and verify zeros using graphical method.

Answer to Problem 30E

  $-5i,5i,-2i,2i$

Explanation of Solution

Given:

Function: $f(x)=x^4+29x^2+100$

Calculation:

  $\begin{array}{l}\Rightarrow f(x)=x^4+29x^2+100\\ \Rightarrow f(x)=x^4+25x^2+4x^2+100\\ \Rightarrow f(x)=x^2(x^2+25)+4(x^2+25)\\ \Rightarrow f(x)=(x^2+25)(x^2+4)\end{array}$

To find the roots of given function, put $f(x)=0,$

  $\begin{array}{l}\Rightarrow (x^2+25)(x^2+4)=0\\ \\ \begin{array}{cc}\begin{array}{l}\Rightarrow x^2+25=0\\ \Rightarrow x^2=-25\\ \Rightarrow x=\pm \sqrt{-25}\\ \Rightarrow x=\pm 5i\\ \Rightarrow x=5i,-5i\end{array}& \begin{array}{l}\Rightarrow x^2+4=0\\ \Rightarrow x^2=-4\\ \Rightarrow x=\pm \sqrt{-4}\\ \Rightarrow x=\pm 2i\\ \Rightarrow x=2i,-2i\end{array}\end{array}\\ \\ \Rightarrow x=-5i,5i,-2i,2i\end{array}$

Now, writing the given function as product of linear factors,

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

As the zeros of the function are imaginary zeros, verification of roots through graphical method is not possible.

Conclusion:

Therefore, the zeros of given function are $x=-5i,5i,-2i\text{and}2i.$