Factor each polynomial. See Examples 5 and 6. (a+b)^2-16

Verified step by step guidance

1. Identify the expression as a difference of squares: \((a+b)^2 - 16\).
2. Recall the difference of squares formula: \(x^2 - y^2 = (x-y)(x+y)\).
3. Recognize that \((a+b)^2\) is \(x^2\) and \(16\) is \(y^2\), where \(y = 4\).
4. Apply the difference of squares formula: \((a+b-4)(a+b+4)\).
5. Verify the factorization by expanding \((a+b-4)(a+b+4)\) to ensure it equals the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions and solving equations. Common techniques include identifying common factors, using special product formulas, and applying methods like grouping or the quadratic formula.

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 (a + b)^2 - 16, recognizing it as a difference of squares allows for efficient factoring, as 16 can be expressed as 4^2.

Special Product Formulas

Special product formulas are algebraic identities that simplify the multiplication of binomials. For example, (a + b)^2 = a^2 + 2ab + b^2 is a key identity. Understanding these formulas helps in recognizing patterns in polynomials, making it easier to factor expressions like (a + b)^2 - 16 by first expanding or simplifying them.