How do you factor #f(x)=x^3+3x^2-12x-18#?
To factorize a n-degree-polynomial function, one must discover its roots that will allow to rearrange the function in this way
For a polynomial of the third degree, when all coefficients are real numbers, there's at least one real root. The other two roots are either real numbers or complex conjugate numbers.
For a general solution of roots of the third degree, requiring some ability to work with complex numbers, I recommend this source:
Roots of a cubic function
But we can try to resolve the problem in the easy way.
The coefficient d may help us to find at least one real root.
This is why we should try the numbers 1, 2, 3, 6, 9, 18 and 1/2, 1/3, 1/6, 1/9 and 1/18 to verify if at least one of them is a root of the function.
In the present case, we discover that 3 is a root (
Then we should divide the polynomial by
The second degree factor can still be further factorized:
Then the function factorized is