In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x⁴−162

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Identify the greatest common factor (GCF) of the terms in the polynomial. In this case, the GCF is 2. Factor out the GCF: 2(x^4 - 81).

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

Rewrite x^4 - 81 as (x^2)^2 - 9^2, where x^2 is the square of x and 9 is the square of 3. This allows us to apply the difference of squares formula.

Factor x^4 - 81 using the difference of squares formula: x^4 - 81 = (x^2 - 9)(x^2 + 9).

Notice that x^2 - 9 is itself a difference of squares. Factor it further using the same formula: x^2 - 9 = (x - 3)(x + 3). The final factored form of the polynomial is 2(x - 3)(x + 3)(x^2 + 9).

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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 is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor, using special products like the difference of squares, and applying the quadratic formula for polynomials of degree two.

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Difference of Squares

The difference of squares is a specific factoring technique applicable to expressions of the form a^2 - b^2, which 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 greatest common factor, allowing for further simplification. This concept is crucial for efficiently factoring certain types of polynomials.

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Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. Identifying the GCF is often the first step in factoring polynomials, as it simplifies the expression and makes it easier to apply other factoring techniques. In the polynomial 2x^4 - 162, recognizing and factoring out the GCF is essential for simplifying the expression before applying further factoring methods.

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