5Question: Let $ f(x) $ be a polynomial such that $ f(x^2 + 2) = x^4 + 4x^2 + 4 $. Find $ f(x^2 - 2) $.

February 22, 2026

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.

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) $.

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:

$$ x^4 + 4x^2 + 4 = (x^2 + 2)^2 $$

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 $:

$$ f(x^2 - 2) = (x^2 - 2)^2 $$

Expand the square:

$$ (x^2 - 2)^2 = x^4 - 4x^2 + 4 $$

Step 3: Final Answer

Thus, the desired expression is:

$$ f(x^2 - 2) = x^4 - 4x^2 + 4 $$

Why This Method Works

By identifying that $ f(x^2 + 2) = (x^2 + 2)^2 $, we leveraged substitution to deduce the closed-form of $ f $. This technique—using functional form and polynomial identity—is a cornerstone in solving such problems efficiently.

SEO Keywords & Phrases

polynomial functional equation
find $ f(x^2 - 2) $ given $ f(x^2 + 2) = x^4 + 4x^2 + 4 $
solve $ f(x^2 + 2) = (x^2 + 2)^2 $
algebraic substitution in polynomials
determine polynomial from functional relation

Conclusion

Polynomial functional equations can be cracked with careful observation and substitution. Here, recognizing the perfect square transformed a seemingly complex equation into a straightforward identification. From $ f(x) = x^2 $, computing $ f(x^2 - 2) $ yields $ x^4 - 4x^2 + 4 $. This elegant solution illustrates how foundational algebraic reasoning powers advanced problem solving.

Ready to explore more polynomial puzzles? Try substituting other expressions like $ f(x^2 - 2) $ with different inputs to deepen your understanding.