# Polynomial Problem f(x^2+2)=x^4+10x^2+4...

1. Aug 21, 2015

### Dapperdub

so i transferred to a new school and i'm collaborating with another pre-cal teacher. she is kind of helpful but i can tell she doesn't really want to share her work (even though collaboration is about helping each other out).

She already made a unit test, but i had to make my own answer key. I got stuck with the final problem

Let f be a polynomial function such that, for all real x,
f(x^2+2)=x^4+10x^2+4. For all real x, what is f(x^2-2)?

The only thing I can think of is

f(x^2+2)= f(x^2)+f(2)

sadly this is all i can come up with. Any website or youtube link would be grateful to help me in the right direction. Thx

2. Aug 21, 2015

### PeroK

Your idea is clearly not correct. Polynomials are not linear, so don't work like that.

Instead, can you see what sort of polynomial $f$ must be?

3. Aug 21, 2015

### Dapperdub

An even function with a base of quadratic? Im not sure how to approch it whatsoever, that's why I'm asking for help.

4. Aug 21, 2015

### PeroK

Does that mean you think $f$ might be a quadratic?

5. Aug 21, 2015

### ehild

You have a function f(g(x)) = x4+10x2+4, with g(x)=x2+2. (Think of composition of functions) What can be the form of the function f(g)?

6. Aug 21, 2015

### HallsofIvy

Staff Emeritus
You are given that $f(x^2+ 2)= x^4+ 10x+ 4$. Let $y= x^2+ 2$ so that $x^2= y- 2$ and $x^4= (y- 2)^2$.

Now, what is f(y)? What do you get when you replace y with x- 2?

7. Aug 21, 2015

### ehild

You meant replacing y by x2-2

8. Aug 21, 2015

### HallsofIvy

Staff Emeritus
Yes, thank you.