Question

Given the expression: 6x10 − 96x2

Part A: Rewrite the expression by factoring out the greatest common factor. (4 points)

Part B: Factor the entire expression completely. Show the steps of your work. (6 points)

Answer 1

The given expression is factored by first finding the greatest common factor, which is 6x^2, to get 6x^2(x^8 - 16). Then, the expression is factored completely as 6x^2(x^4 + 4)(x^2 + 2)(x^2 - 2), using the difference of squares technique.

Step-by-step explanation:

Let's address the given expression in two parts, as requested:

Part A: Factoring out the Greatest Common Factor

The expression given is: 6x^{10} − 96x^2.

To factor out the greatest common factor (GCF), we should identify the highest power of x that is common to both terms, and the largest number that can divide both coefficients. Here, the GCF is 6x^2. Thus, the expression factored by the GCF is:

6x^2(x^8 − 16).

Part B: Factoring Completely

Now we have the expression: 6x^2(x^8 − 16). Notice that x^8 − 16 is a difference of squares, which can be factored as (x^4 + 4)(x^4 − 4). The term x^4 − 4 is also a difference of squares and can be further factored to (x^2 + 2)(x^2 − 2). Hence, the entire expression factored completely is:

6x^2(x^4 + 4)(x^2 + 2)(x^2 − 2).