Question

`f(x) = (x ^(2) + 4)(x - 2)`find all real and non-real roots of the function

Answer 1

Given the following function

`f(x)=(x^2+4)(x-2)`

We want to find the roots for this function. The roots are given as the solutions for the following equation:

`(x^2+4)(x-2)=0`

Since it is already factorized, we can use the property that the product of two terms is zero if and only if at least one of them is zero. Then, the solutions of this equation are the same solutions for this system:

`\begin{cases}x^2+4=0 \\ x-2=0\end{cases}`

Solving the first equation we have:

`\begin{gathered} x^2+4=0 \\ x^2=-4 \\ x=\pm\sqrt[]{-4}=\pm\sqrt[]{(-1)(4)}=\pm\sqrt[]{(-1)}\sqrt[]{4} \\ x=\pm2i \end{gathered}`

Now, solving the second equation

`x-2=0\Rightarrow x=2`

Then, our solutions are:

`\begin{gathered} x_1=2i \\ x_2=-2i \\ x_3=2 \end{gathered}`

This corresponds to the option B.