Sure, let's tackle this step by step.
### Part A: Factoring out the Greatest Common Factor (GCF)
We start with the given expression:
[tex]\[ 6 x^{10} - 96 x^2 \][/tex]
The first step is to determine the greatest common factor (GCF) of the terms. Both terms, [tex]\(6 x^{10}\)[/tex] and [tex]\(96 x^2\)[/tex], share a common factor of [tex]\(6 x^2\)[/tex].
To factor out the GCF, we divide each term by [tex]\(6 x^2\)[/tex]:
[tex]\[ 6 x^{10} \div 6 x^2 = x^8 \][/tex]
[tex]\[ 96 x^2 \div 6 x^2 = 16 \][/tex]
Thus, we can rewrite the expression as:
[tex]\[ 6 x^2 (x^8 - 16) \][/tex]
So, for Part A, the factored expression after factoring out the greatest common factor is:
[tex]\[ 6x^2(x^8 - 16) \][/tex]
### Part B: Factoring the Entire Expression Completely
We continue from the expression obtained in Part A:
[tex]\[ 6 x^2 (x^8 - 16) \][/tex]
Next, we notice that [tex]\(x^8 - 16\)[/tex] is a difference of squares. Recall the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In our case, we can write [tex]\(x^8\)[/tex] as [tex]\((x^4)^2\)[/tex] and 16 as [tex]\(4^2\)[/tex]. Thus:
[tex]\[ x^8 - 16 = (x^4)^2 - 4^2 = (x^4 - 4)(x^4 + 4) \][/tex]
So now our expression becomes:
[tex]\[ 6 x^2 (x^4 - 4)(x^4 + 4) \][/tex]
Next, we look at [tex]\(x^4 - 4\)[/tex]. This is again a difference of squares:
[tex]\[ x^4 - 4 = (x^2)^2 - 2^2 = (x^2 - 2)(x^2 + 2) \][/tex]
Therefore, the completely factored form of the expression is:
[tex]\[ 6 x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
Thus, for Part B, the completely factored expression is:
[tex]\[ 6 x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
#### Summary:
- Part A: The expression factored by the GCF is [tex]\(6x^2(x^8 - 16)\)[/tex].
- Part B: The completely factored expression is [tex]\(6x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4)\)[/tex].
These steps detail the process of fully factoring the given polynomial expression.
