Factoring Equations Completely?

Question

The directions say factor the equation completely using the rational zero theorem and synthetic division, and solve for all values of x.

x^4 - 2x^3 - 13x^2 + 14x + 24 = 0

Thanks in advance. I usually can solve these types of problems, but I am a bit rusty.

Answer 1

Try factors of 24 to determine if there is a binomial (x + a) that will be a factor of your equation.  The rational zero theorem means find 'a' such that:

f(a) = a^4 - 2a^3 - 13a^2 + 14a + 24 = 0

Use synthetic division when you find the binomial factor.

When you have found a binomial factor, synthetic division will result in a new polynomial of the third degree.  Repeat process to find another binomial factor.  Then you will have a quadratic.  You know how to factor a quadratic---yes?

This factors to:

(x - 2)(x + 3)(x - 4)(x + 1) = 0

(You can check by multiplying!)

The solutions (zeroes) are:  x = 2, -3, 4, -1

Check.