Given the expression: 6x10 − 96x2Part 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)

Part A: Rewrite the expression by factoring out the greatest common factor.

First, identify the greatest common factor (GCF) of the terms in the expression 6x^10 and -96x^2.

• The GCF of the coefficients 6 and -96 is 6.
• The GCF of the variables x^10 and x^2 is x^2.

Therefore, the GCF of the entire expression is 6x^2.

Now, factor out the GCF from each term:

6x^10 − 96x^2 = 6x^2(x^8) − 6x^2(16)

So, the expression factored by the GCF is:

6x^10 − 96x^2 = 6x^2(x^8 − 16)

Part B: Factor the entire expression completely.

Now, we need to factor the expression inside the parentheses completely: x^8 − 16.

Notice that x^8 − 16 is a difference of squares, which can be factored using the formula a^2 − b^2 = (a − b)(a + b).

Here, a = x^4 and b = 4, so:

x^8 − 16 = (x^4)^2 − 4^2 = (x^4 − 4)(x^4 + 4)

Next, we need to check if (x^4 − 4) and (x^4 + 4) can be factored further.

• (x^4 − 4) is also a difference of squares: (x^4 − 4) = (x^2)^2 − 2^2 = (x^2 − 2)(x^2 + 2)
• (x^4 + 4) is a sum of squares and cannot be factored further over the real numbers.

So, the completely factored form of the expression is:

6x^2(x^8 − 16) = 6x^2(x^4 − 4)(x^4 + 4) = 6x^2(x^2 − 2)(x^2 + 2)(x^4 + 4)

Therefore, the final factored expression is:

6x^2(x^2 − 2)(x^2 + 2)(x^4 + 4)