Question

Find any real zeros of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer 1

We want to find real roots for the following function

`f(x)=6(x^2+6)(x^2+8)^2`

The roots of this function are the solutions for the following equation

`6(x^2+6)(x^2+8)^2=0`

Since we have a product of two terms equals zero, this means that at least one of them is zero.

`\begin{gathered} 6(x^2+6)(x^2+8)^2=0 \\ (x^2+6)(x^2+8)^2=0\Rightarrow\begin{cases}(x^2+6)=0 \\ (x^2+8)^2=0\end{cases} \end{gathered}`

Solving those two equations:

`\begin{cases}(x^2+6)=0 \\ (x^2+8)^2=0\end{cases}\Rightarrow\begin{cases}x^2=-6 \\ x^2=-8\end{cases}\Rightarrow\begin{cases}x^{}=\pm\sqrt[]{6}i \\ x^{}=\pm2\sqrt[]{2}i\end{cases}`

This function doesn't have real solutions.