In Exercises 1–68, factor completely, or state that the polynomial is prime. 4x² + 25y²

Verified step by step guidance

1. Identify the form of the polynomial: The given polynomial is 4x^2 + 25y^2.
2. Recognize that this is a sum of squares: a^2 + b^2, where a = 2x and b = 5y.
3. Recall that a sum of squares cannot be factored over the real numbers into linear factors.
4. Check if there are any common factors: In this case, there are no common factors between the terms 4x^2 and 25y^2.
5. Conclude that the polynomial is prime over the real numbers, as it cannot be factored further.

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common techniques include finding common factors, using the difference of squares, and applying special products like perfect square trinomials.

Sum of Squares

The sum of squares refers to an expression of the form a² + b², which cannot be factored over the real numbers. In the case of the polynomial 4x² + 25y², it represents a sum of squares where 4x² is (2x)² and 25y² is (5y)². Recognizing this helps determine that the polynomial is prime.

Prime Polynomials

A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with real coefficients. Identifying a polynomial as prime is crucial in algebra, as it indicates that no further simplification is possible. In this case, since 4x² + 25y² is a sum of squares, it is classified as prime.