Question

How do you factor completely `x^3 + 4x^2 + 5x + 2`?

Answer 1

`(x+1)^2(x+2)`

Explanation:

The factorization of a polynom of degree 3 is difficult. . There is a formula equivalent to the one of second degree polynoms, but it is very difficult to use. The only way you have is to solve it numerically, or trying numbers by chance. You have turned the solution easy, since -1 is an obvious solution.

To find the other roots we need to divide `x^3+4x^2+5x+2` by `x+1`:

`(x^3+4x^2+5x+2)/(x+1)`

=`x^2+(3x^2+5x+2)/(x+1)`

=`x^2+3x+ (2x+2)/(x+1)`

=`x^2+3x+2`

Now we can apply thye general formula for a second degree polynom:

`x=(-3+-sqrt(3^2-4*2))/2`

`x=(-3+-sqrt(1))/2`

`x=(-3+-1)/2`

`x=(-4)/2 or x=(-2)/2`

`x=-2 or x=-1`

So the factorization will be

the product of (x-roots) or

`(x+1)^2(x+2)`