How would I factorize
x^3+2x^2-9x-18
and then solve
f(x)=0?
Thanks a lot.
Factor by grouping: (Typically if you are asked to factor a cubic it will be a variation of this method.)
$\displaystyle (x^3 + 2x^2) - (9x + 18)$
$\displaystyle x^2(x + 2) - 9(x + 2)$
$\displaystyle (x^2 - 9)(x + 2)$
Now note that $\displaystyle x^2 - 9$ factors:
So we get:
$\displaystyle (x + 3)(x - 3)(x + 2)$
So if we have $\displaystyle f(x) = x^3+2x^2-9x-18 = 0$
Then
$\displaystyle (x + 3)(x - 3)(x + 2) = 0$
So x = -3, -2, or 3.
-Dan
also if you do not know how to factor that well you could try and find one root, with the rational root theorem.
lets say that u have $\displaystyle f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$ now since this has a root lets call it $\displaystyle r$ we then have
$\displaystyle f(x)=(x-r)Q(x)$
where $\displaystyle Q(x)=y_nx^{n-1}+y{n-1}x^{n-1}+\dots+y_2x+y_1$. as you notice the degree of Q is one less than that of f.