Question

If f(x)=2x^3+14x^2+30x+50 and f(-5)=0, then find all of the zeros of f(x).

Answer 1

Since f(-5) = 0, that means x = -5 is a zero of the function.

To find the other zeros, let's divide f(x) by x+5, this way we can reduce the polynomial to a quadratic function and then find the other zeros.

So we have:

The resulting polynomial is 2x² + 4x + 10.

Calculating the zeros with the quadratic formula, we have:

$\begin{gathered} a=2,b=4,c=10 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_1=\frac{-4+\sqrt[]{16-80}}{4}=\frac{-4+8i}{4}=-1+2i \\ x_2=\frac{-4-8i}{4}=-1-2i \end{gathered}$

Therefore the zeros of f(x) are:

-5, -1 + 2i, -1 - 2i.