In Exercises 65–92, factor completely, or state that the polynomial is prime. 9x³−9x

Verified step by step guidance

1. Step 1: Identify the greatest common factor (GCF) of the terms in the polynomial. The terms in the polynomial are 9x³ and -9x. The GCF is 9x because both terms share a factor of 9 and at least one factor of x.
2. Step 2: Factor out the GCF (9x) from the polynomial. This means dividing each term by 9x and writing the polynomial as a product of the GCF and the resulting expression. The polynomial becomes 9x(x² - 1).
3. Step 3: Recognize that the expression inside the parentheses, x² - 1, is a difference of squares. Recall the formula for factoring a difference of squares: a² - b² = (a - b)(a + b).
4. Step 4: Apply the difference of squares formula to x² - 1. Here, a = x and b = 1, so x² - 1 factors into (x - 1)(x + 1).
5. Step 5: Write the fully factored form of the polynomial. Combine the GCF and the factored expression to get 9x(x - 1)(x + 1).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying the quadratic formula for polynomials of degree two.

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. In polynomial expressions, identifying the GCF allows for simplification by factoring it out, which can make the remaining polynomial easier to work with. For example, in the polynomial 9x^3 - 9x, the GCF is 9x.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is crucial in algebra, as it indicates that the polynomial cannot be simplified further. In the case of 9x^3 - 9x, after factoring out the GCF, the remaining polynomial can be analyzed to determine if it is prime.