Factor each polynomial. See Examples 5 and 6. 9a²-16

Verified step by step guidance

1. Recognize that the polynomial \$9a^2 - 16\( is a difference of squares because it can be written as \)(3a)^2 - 4^2$.
2. Recall the difference of squares factoring formula: \(x^2 - y^2 = (x - y)(x + y)\).
3. Identify \(x = 3a\) and \(y = 4\) in the expression \$9a^2 - 16$.
4. Apply the formula to factor the polynomial as \((3a - 4)(3a + 4)\).
5. Write the final factored form: \((3a - 4)(3a + 4)\).

Key Concepts

Difference of Squares

The difference of squares is a special factoring pattern where an expression is written as a^2 - b^2. It factors into (a - b)(a + b). Recognizing this pattern helps simplify polynomials like 9a^2 - 16 by identifying perfect squares and applying the formula.

Perfect Squares

Perfect squares are numbers or expressions that are squares of integers or variables, such as 9a^2 (which is (3a)^2) and 16 (which is 4^2). Identifying perfect squares is essential to apply the difference of squares factoring method correctly.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process simplifies expressions and solves equations. Recognizing patterns like difference of squares is a key step in factoring efficiently.