In Exercises 65–92, factor completely, or state that the polynomial is prime. x²−10x+25−36y²

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1. Identify the structure of the polynomial: The given expression is \( x^2 - 10x + 25 - 36y^2 \). Notice that \( x^2 - 10x + 25 \) is a quadratic trinomial, and \( 36y^2 \) is a perfect square.
2. Factor \( x^2 - 10x + 25 \): Recognize that \( x^2 - 10x + 25 \) is a perfect square trinomial. It can be factored as \( (x - 5)^2 \), since \( (x - 5)(x - 5) = x^2 - 10x + 25 \).
3. Rewrite the expression: Substitute \( (x - 5)^2 \) for \( x^2 - 10x + 25 \). The expression becomes \( (x - 5)^2 - 36y^2 \).
4. Recognize the difference of squares: The expression \( (x - 5)^2 - 36y^2 \) is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = (x - 5) \) and \( b = 6y \).
5. Factor the difference of squares: Apply the formula to factor \( (x - 5)^2 - 36y^2 \) as \( ((x - 5) - 6y)((x - 5) + 6y) \).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. Common techniques include identifying common factors, using the difference of squares, and applying the quadratic formula when necessary.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions of the form a^2 - b^2, which can be factored into (a - b)(a + b). In the given polynomial, recognizing the structure of a difference of squares is crucial for simplifying the expression effectively.

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically written in the form ax^2 + bx + c. Understanding the properties of quadratics, including their standard form and how to complete the square, is vital for factoring and solving quadratic equations. In this case, recognizing the quadratic nature of the expression helps in the factoring process.