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

[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 problemLet 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

 

[PeroK]

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 problemLet 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

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?

 

[Dapperdub]

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

 

[PeroK]

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

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

 

[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)?

 

[HallsofIvy]

You are given that [itex]f(x^2+ 2)= x^4+ 10x+ 4[/itex]. Let [itex]y= x^2+ 2[/itex] so that [itex]x^2= y- 2[/itex] and [itex]x^4= (y- 2)^2[/itex].

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

 

[ehild]

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

You meant replacing y by x²-2

 

[HallsofIvy]

Yes, thank you.