In Exercises 39–48, factor the difference of two squares. 16x^4−81

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic expression that takes the form a^2 - b^2, which can be factored into (a - b)(a + b). This concept is fundamental in algebra as it simplifies expressions and solves equations efficiently. In the given problem, recognizing that 16x^4 and 81 are both perfect squares allows us to apply this factoring technique.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer or a variable. For example, 16x^4 is a perfect square because it can be written as (4x^2)^2, and 81 is a perfect square since it equals 9^2. Identifying perfect squares is crucial for factoring expressions like the one in the question, as it enables the application of the difference of squares formula.

Factoring Techniques

Factoring techniques involve breaking down expressions into simpler components that, when multiplied together, yield the original expression. This process is essential in algebra for simplifying equations, solving polynomial equations, and analyzing functions. In this case, applying the difference of squares technique allows us to factor the expression 16x^4 - 81 into (4x^2 - 9)(4x^2 + 9), demonstrating the utility of these techniques.