Factor each polynomial. See Examples 5 and 6. x^4-16

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 techniques include identifying common factors, using special product formulas, and applying methods like grouping or synthetic division.

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 case of the polynomial x^4 - 16, it can be recognized as a difference of squares since 16 is 4^2, allowing for further factoring into simpler components.

Quadratic Form

A polynomial can sometimes be expressed in a quadratic form, which is a polynomial of degree two. In the case of x^4 - 16, after applying the difference of squares, it can be factored into (x^2 - 4)(x^2 + 4). The term x^2 - 4 can be further factored as another difference of squares, illustrating the importance of recognizing quadratic structures in higher-degree polynomials.