Factoring help for functions

[Rose Lily]

Hi!

I have two functions that I've determined are even, odd or neither. Now I need to solve them algebraically, but it's been a few months since I've factored (I'm in a college algebra 111 class) (took a term off to fufill other class requirements) and in that time I have forgotten more than I though was posssible!! Ugg. Below are the functions.

f(x) = x^3 - 2x
TEST: f(-x) = (-x)^3 -2(-x)
=-x^3 + 2x
Signs change, ODD

However, how do I factor this? I see the GCF of x, which gives me is x(x^2 - 2) but what do I do after that? Please show me step by step.

Then, there's f(x) = 3x - x^2
TEST: f(-x) = 3(-x) - (-x)^2
=-3x - x^2
Signs different, neither.

Factoring f(x) =3x - x^2
GCF
x(3-x)

After that?

 

[JeffM]

Hi!

I have two functions that I've determined are even, odd or neither. Now I need to solve them algebraically, but it's been a few months since I've factored (I'm in a college algebra 111 class) (took a term off to fufill other class requirements) and in that time I have forgotten more than I though was posssible!! Ugg. Below are the functions.

f(x) = x^3 - 2x
TEST: f(-x) = (-x)^3 -2(-x)
=-x^3 + 2x
Signs change, ODD

However, how do I factor this? I see the GCF of x, which gives me is x(x^2 - 2) but what do I do after that? Please show me step by step.

Do you understand that factoring is a way to solve certain kinds of problems, and how you factor depends on the problem? In this case, if you are trying to factor with rational numbers x(x^2 - 2) is the best that you can do. If, however, you are factoring over the real numbers, you can factor further if you see that the term (x^2 - 2) can be treated as a difference of squares as follows

\(\displaystyle x^3 - 2x = x(x^2 - 2) = x\left\{x^2 - \left(\sqrt{2}\right))^2\right\} = x(x - \sqrt{2})(x + \sqrt{2}).\)

Then, there's f(x) = 3x - x^2
TEST: f(-x) = 3(-x) - (-x)^2
=-3x - x^2
Signs different, neither.

Factoring f(x) =3x - x^2
GCF
x(3-x) You COULD factor (3 - x) further by using square roots, but why WOULD you?

After that?

Does this help?