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

Verified step by step guidance

1. Recognize that the given expression \( (a+b)^2 - 16 \) is a difference of squares, since \$16\( can be written as \)4^2$.
2. Recall the difference of squares formula: \(x^2 - y^2 = (x - y)(x + y)\).
3. Identify \(x\) as \((a+b)\) and \(y\) as \$4\( in the expression \) (a+b)^2 - 4^2 $.
4. Apply the difference of squares formula: \(((a+b) - 4)((a+b) + 4)\).
5. Simplify each binomial to get the factored form: \((a + b - 4)(a + b + 4)\).

Key Concepts

Difference of Squares

The difference of squares is a factoring technique used when an expression is in the form a² - b². It factors into (a - b)(a + b). Recognizing this pattern helps simplify expressions like (a+b)² - 16, where 16 is a perfect square.

Expanding and Recognizing Perfect Squares

Understanding how to expand and recognize perfect square trinomials, such as (a+b)² = a² + 2ab + b², is essential. This knowledge helps identify when an expression can be rewritten or factored using special formulas.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting them as products of simpler polynomials. Techniques include factoring out the greatest common factor, grouping, and applying special formulas like difference of squares or perfect square trinomials.