5Question: Let $ f(x) $ be a polynomial such that $ f(x^2 + 2) = x^4 + 4x^2 + 4 $. Find $ f(x^2 - 2) $. - All Square Golf
Title: Solving Functional Polynomial Equations: Find $ f(x^2 - 2) $ Given $ f(x^2 + 2) = x^4 + 4x^2 + 4 $
Title: Solving Functional Polynomial Equations: Find $ f(x^2 - 2) $ Given $ f(x^2 + 2) = x^4 + 4x^2 + 4 $
Meta Description:
Explore algebraic reasoning and polynomial substitution with $ f(x^2 + 2) = x^4 + 4x^2 + 4 $. Learn how to determine $ f(x^2 - 2) $ step-by-step.
Understanding the Context
Introduction
Functional equations involving polynomials often reveal deep structure when approached systematically. One such problem asks:
> Let $ f(x) $ be a polynomial such that $ f(x^2 + 2) = x^4 + 4x^2 + 4 $. Find $ f(x^2 - 2) $.
At first glance, this may seem abstract, but by applying substitution and polynomial identification, we unlock a clear path forward. This article guides you through solving this elegant functional equation and computing $ f(x^2 - 2) $.
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Key Insights
Step 1: Analyze the Given Functional Equation
We are given:
$$
f(x^2 + 2) = x^4 + 4x^2 + 4
$$
Notice that the right-hand side is a perfect square:
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$$
x^4 + 4x^2 + 4 = (x^2 + 2)^2
$$
So the equation becomes:
$$
f(x^2 + 2) = (x^2 + 2)^2
$$
This suggests that $ f(u) = u^2 $, where $ u = x^2 + 2 $. Since this holds for infinitely many values (and both sides are polynomials), we conclude:
$$
f(u) = u^2
$$
That is, $ f(x) = x^2 $ is the polynomial satisfying the condition.
Step 2: Compute $ f(x^2 - 2) $
Now that we know $ f(u) = u^2 $, substitute $ u = x^2 - 2 $:
$$
f(x^2 - 2) = (x^2 - 2)^2
$$
