Sure, let's tackle the problem step-by-step.
### Given Expression
[tex]\[ 3x^{10} - 48x^2 \][/tex]
### Part A: Factoring Out the Greatest Common Factor (GCF)
To find the greatest common factor (GCF), let's look at both terms:
1. The first term is [tex]\(3x^{10}\)[/tex].
2. The second term is [tex]\(-48x^2\)[/tex].
The GCF of the coefficients [tex]\(3\)[/tex] and [tex]\(-48\)[/tex] is [tex]\(3\)[/tex].
For the variable parts, the GCF of [tex]\(x^{10}\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex].
Thus, the GCF of the entire expression is [tex]\(3x^2\)[/tex].
Now factor out [tex]\(3x^2\)[/tex] from the given expression:
[tex]\[ 3x^{10} - 48x^2 = 3x^2 (x^8 - 16) \][/tex]
So, after factoring out the GCF, we get:
[tex]\[ 3x^{10} - 48x^2 = 3x^2 (x^8 - 16) \][/tex]
### Part B: Factoring the Entire Expression Completely
Next, we need to factor the expression [tex]\(x^8 - 16\)[/tex] entirely.
Notice that [tex]\(x^8 - 16\)[/tex] is a difference of squares:
[tex]\[ x^8 - 16 = (x^4)^2 - (4)^2 = (x^4 - 4)(x^4 + 4) \][/tex]
Now, let's focus on factoring [tex]\(x^4 - 4\)[/tex] and [tex]\(x^4 + 4\)[/tex].
#### Factoring [tex]\(x^4 - 4\)[/tex]
Again, [tex]\(x^4 - 4\)[/tex] is a difference of squares:
[tex]\[ x^4 - 4 = (x^2)^2 - (2)^2 = (x^2 - 2)(x^2 + 2) \][/tex]
#### Factoring [tex]\(x^4 + 4\)[/tex]
[tex]\(x^4 + 4\)[/tex] is not a difference of squares and requires factoring as a sum of squares which can be broken down further:
First, we use the sum of squares formula:
[tex]\[ x^4 + 4 = x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2) \][/tex]
Thus, the complete factorization of [tex]\(x^8 - 16\)[/tex] is:
[tex]\[ x^8 - 16 = (x^2 - 2)(x^2 + 2)(x^2 - 2x + 2)(x^2 + 2x + 2) \][/tex]
Combining all parts together:
The fully factored form of the original expression [tex]\(3x^{10} - 48x^2\)[/tex] is:
[tex]\[ 3x^{10} - 48x^2 = 3x^2 (x^2 - 2)(x^2 + 2)(x^2 - 2x + 2)(x^2 + 2x + 2) \][/tex]
So the complete factorization is:
[tex]\[ 3x^2 (x^2 - 2)(x^2 + 2)(x^2 - 2x + 2)(x^2 + 2x + 2) \][/tex]
And that completes the problem!
