Question

How do you factor completely `x^3 - 8x^2 + 5x + 50 = 0`?

Answer 1

`(x+2)(x-5)^2`.

Explanation:

We know that if a number solves such an equation, it must divide its last coefficient, i.e. `50`. So, we can try with some of its divisors: if `f(x)=x^3-8x^2+5x+50`, then:

• `f(1)=48 ne 0`;
• `f(-1)= 36 ne 0`
• `f(2) =36 ne 0`
• `f(-2) = 0`.

So, `-2` is a solution, which means that `f(x)` can be divided by `(x+2)`. Do the long division, and you have that

`x^3-8x^2+5x+50 = (x+2)(x^2-10x+25)`

Now we must factor the second parenthesis, but you can see that it is the square of `(x-5)`, and so `f(x)` is completely factored.