Exercise 1820Algebra › PolynomialsL1Added 3 Nov 2026

Decomposition into irreducible factors

#factorization#irreducible factors#real polynomials

Statement

Factor $P(X)=X^4+4$ over $\mathbb{R}$ .

Hint

Use Sophie Germain's identity: $a^4+4b^4=(a^2+2ab+2b^2)(a^2-2ab+2b^2)$ .

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Sophie Germain. With $a=X$ , $b=1$ :
$X^4+4=(X^2+2X+2)(X^2-2X+2).$

Irreducibility. Each quadratic has negative discriminant: $\Delta=4-8=-4<0$ and $\Delta=4-8=-4<0$ . So both are irreducible over $\mathbb{R}$ .

Conclusion. $P(X)=(X^2+2X+2)(X^2-2X+2)$ , both factors irreducible over $\mathbb{R}$ .