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

  • Thread starter Dapperdub
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In summary: Apologies for the typo. In summary, the conversation discusses collaboration between two pre-calculus teachers and a problem involving a polynomial function. The function is given as f(x^2+2)=x^4+10x^2+4 and the task is to find f(x^2-2). The conversation also explores different approaches and ideas for solving the problem.
  • #1
Dapperdub
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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
 
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  • #2
Dapperdub said:
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?
 
  • #3
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.
 
  • #4
Dapperdub said:
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?
 
  • #5
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
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?
 
  • #7
HallsofIvy said:
Now, what is f(y)? What do you get when you replace y with x- 2?

You meant replacing y by x2-2 :oldsmile:
 
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Likes HallsofIvy
  • #8
Yes, thank you.
 

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

1. What is a polynomial problem?

A polynomial problem involves solving equations or expressions that contain variables raised to different powers. These powers are called exponents and they can be any whole number, including negative numbers.

2. How do you solve a polynomial problem?

To solve a polynomial problem, you need to set the expression equal to zero and use algebraic techniques such as factoring, the quadratic formula, or synthetic division to find the solutions.

3. What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial problem f(x^2+2)=x^4+10x^2+4, the degree is 4 because x is raised to the 4th power.

4. What is the purpose of the polynomial problem f(x^2+2)=x^4+10x^2+4?

The purpose of this polynomial problem is to find the values of x that make the equation true. These values are called solutions or roots of the equation.

5. What is the importance of polynomial problems in science?

Polynomial problems are used in various fields of science, such as physics, chemistry, and engineering, to model and solve real-world problems. They are also important in data analysis and computer science for curve fitting and data interpolation.

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