Question

Factor the polynomial function f(x). Then solve the equation f(x)=0.

Answer 1

$f(x)=x^3+13x^2+34x-48$

We can immediately check that f(1) = 0. Then we can divide the polynomial function by (x - 1):

x³ + 13x² + 34x - 48 | x - 1

-x³ + x² | x²

14x² + 34x - 48 | x - 1

-14x² + 14x | x² + 14x

48x - 48 | x - 1

-48x + 48 | x² + 14x + 48

Then we have:

$f(x)=(x-1)\cdot(x^2+14x+48)$

Now, we must find the roots of the quadratic term:

$\begin{gathered} x^2+14x+48=0 \\ x_+=\frac{-14+\sqrt{14^2-4\cdot48}}{2}=-6 \\ x_-=\frac{-14-\sqrt{14^2-4\cdot48}}{2}=-8 \end{gathered}$

Therefore, we have:

$f(x)=(x-1)\cdot(x+6)\cdot(x+8)$