Factor each polynomial. See Examples 5 and 6. y^4-81

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 out the greatest common factor, using special products, and applying techniques like grouping.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions in the form a^2 - b^2, which can be factored into (a - b)(a + b). In the case of the polynomial y^4 - 81, it can be recognized as a difference of squares since y^4 is (y^2)^2 and 81 is 9^2, allowing for straightforward factoring.

Quadratic Factors

After applying the difference of squares, the resulting factors may sometimes be quadratic expressions. For instance, in the case of y^4 - 81, after factoring it as (y^2 - 9)(y^2 + 9), the factor y^2 - 9 can be further factored into (y - 3)(y + 3). Understanding how to factor quadratics is crucial for fully simplifying polynomials.