In Exercises 23–48, factor completely, or state that the polynomial is prime. x² + 25y²

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial expression into simpler components, or factors, that when multiplied together yield the original polynomial. This process is essential for simplifying expressions, solving equations, and analyzing polynomial behavior. Common techniques include identifying common factors, using the difference of squares, and applying special factoring formulas.

Sum of Squares

The expression x² + 25y² represents a sum of squares, which is a specific type of polynomial that cannot be factored over the real numbers. Unlike the difference of squares, which can be factored into two binomials, the sum of squares does not have real factors and is considered prime in the context of real number factorization. Understanding this distinction is crucial for determining whether a polynomial can be factored.

Prime Polynomials

A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is important in algebra as it helps in understanding the limits of factorization and the nature of polynomial roots. In the case of x² + 25y², identifying it as prime indicates that it does not have simpler polynomial factors.