In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^4−162

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial expression into simpler components, or factors, that when multiplied together yield the original polynomial. This process often requires identifying common factors, applying special factoring techniques like difference of squares, or using methods such as grouping. Understanding how to factor is essential for simplifying expressions and solving polynomial equations.

Difference of Squares

The difference of squares is a specific factoring technique used when a polynomial is in the form a^2 - b^2. This can be factored into (a - b)(a + b). In the given polynomial, 2x^4 - 162 can be recognized as a difference of squares after factoring out the common factor, allowing for further simplification and factorization.

Prime Polynomials

A prime polynomial is one that cannot be factored into simpler polynomials with real coefficients. Recognizing whether a polynomial is prime is crucial in algebra, as it determines the methods available for solving equations or simplifying expressions. In the context of the given polynomial, determining if it can be factored completely or is prime is a key step in the problem-solving process.