Overview

Sophie Germain is the standard way to factor a sum of two fourth powers when one coefficient is $4$.

Key Ideas

• $x^4 + 4y^4 = (x^2 - 2xy + 2y^2)(x^2 + 2xy + 2y^2)$.
• Think of it as $(x^2 + 2y^2)^2 - (2xy)^2$.
• It often appears in AIME-level factoring.

Core Skills

Confirm the Coefficient

The identity applies only when the coefficient of $y^4$ is $4$. Factor out constants first to match this pattern.

Use Difference of Squares

Recognize $x^4+4y^4$ as $(x^2+2y^2)^2-(2xy)^2$ to factor quickly.

Look for Substitutions

If the expression is not in the exact form, set $x$ or $y$ as a binomial to create the pattern.

Worked Example

Factor $a^4 + 4$.

Write $a^4 + 4 = a^4 + 4\cdot 1^4$ and apply the identity: $(a^2 - 2a + 2)(a^2 + 2a + 2)$.

More Examples

Example 1: Factor a Constant

Factor $16x^4 + 4y^4$.

Factor out $4$ first to match the Sophie Germain pattern, then apply the identity.

Example 2: Binomial Substitution

Factor $(x^2+1)^2 + 4x^2$.

Treat it as $a^4+4b^4$ with $a=x^2+1$ and $b=x$.

Example 3: Integer Check

Factor $81 + 4y^4$.

Write $81 = 3^4$ and apply the identity with $x=3$.

Strategy Checklist

• Check that the coefficient of $y^4$ is $4$ (or can be made $4$).
• Use the difference of squares view for speed.
• Substitute when the expression is a binomial square.

Common Pitfalls

• Forgetting the coefficient $4$ is essential.
• Mistaking it for a difference of squares without the middle step.
• Applying the identity to $x^4+y^4$ without scaling.

Practice Problems