In Exercises 39–48, factor the difference of two squares. 16x⁴−81

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Recognize that the given expression, 16x^4 - 81, is a difference of two squares. Recall the formula for factoring the difference of two squares: a^2 - b^2 = (a - b)(a + b).

Rewrite each term as a square: 16x^4 can be written as (4x^2)^2, and 81 can be written as 9^2.

Substitute these squares into the formula: (4x^2)^2 - 9^2 = (4x^2 - 9)(4x^2 + 9).

Notice that the first factor, 4x^2 - 9, is itself a difference of two squares. Apply the formula again: 4x^2 - 9 = (2x - 3)(2x + 3).

Combine all the factors: The fully factored form of the expression is (2x - 3)(2x + 3)(4x^2 + 9).

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주요 개념

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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 involving the difference of squares.

Factoring Techniques

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Understanding various factoring techniques, including the difference of squares, is essential for solving polynomial equations and simplifying algebraic expressions. Mastery of these techniques enhances problem-solving skills in algebra.

In Exercises 39–48, factor the difference of two squares. 16x4−81