Factor each polynomial. See Examples 5 and 6. 9m^2-n^2-2n-1

Verified step by step guidance

1. Recognize that the polynomial is a quadratic in terms of two variables, \( m \) and \( n \).
2. Notice that the expression \( 9m^2 - n^2 - 2n - 1 \) can be rearranged to group terms: \( 9m^2 - (n^2 + 2n + 1) \).
3. Observe that \( n^2 + 2n + 1 \) is a perfect square trinomial, which can be factored as \((n + 1)^2\).
4. Rewrite the expression using the difference of squares formula: \( 9m^2 - (n + 1)^2 \).
5. Apply the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \), where \( a = 3m \) and \( b = n + 1 \), to factor the expression as \((3m - (n + 1))(3m + (n + 1))\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for simplifying expressions, solving equations, and analyzing polynomial behavior. Common methods include factoring by grouping, using the difference of squares, and applying the quadratic formula when necessary.

Difference of Squares

The difference of squares is a specific factoring technique applicable to expressions of the form a^2 - b^2, which can be factored into (a - b)(a + b). Recognizing this pattern is crucial when dealing with polynomials that contain squared terms, as it allows for quick simplification and solution of equations.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically in the form ax^2 + bx + c. Understanding how to manipulate and factor these expressions is fundamental in algebra, as they frequently arise in various mathematical contexts, including graphing parabolas and solving real-world problems involving area and projectile motion.