Factor each polynomial. See Examples 5 and 6. $36z^2-81y^4$

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial \(36z^2 - 81y^4\). The GCF is the largest expression that divides both terms evenly.
2. Factor out the GCF from the polynomial. This means rewriting the polynomial as the product of the GCF and the remaining terms inside parentheses.
3. Recognize that the remaining expression inside the parentheses is a difference of squares, since it has the form \(a^2 - b^2\) where \(a = \sqrt{\text{first term}}\) and \(b = \sqrt{\text{second term}}\).
4. Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\) to factor the expression inside the parentheses.
5. Write the fully factored form by combining the GCF and the factored difference of squares.

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 polynomials like 36z² - 81y⁴ by identifying perfect squares.

Greatest Common Factor (GCF)

The GCF is the largest factor shared by all terms in a polynomial. Factoring out the GCF simplifies the expression and makes further factoring easier. For example, factoring out 9 from 36z² - 81y⁴ reduces it to 9(4z² - 9y⁴).

Factoring Polynomials

Factoring polynomials involves rewriting them as a product of simpler polynomials. This process often includes factoring out the GCF and applying special formulas like difference of squares. Mastery of these techniques is essential for simplifying and solving polynomial expressions.