Sure, let's work through this step-by-step.
### Part A: Factoring out the Greatest Common Factor (GCF)
The given expression is:
[tex]\[ 4x^{10} - 64x^2 \][/tex]
1. Identify the Greatest Common Factor (GCF):
- The GCF in terms of the numerical part between [tex]\(4\)[/tex] and [tex]\(64\)[/tex] is [tex]\(4\)[/tex].
- The GCF in terms of the powers of [tex]\(x\)[/tex] between [tex]\(x^{10}\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex].
2. Factor out the GCF:
- We can factor out [tex]\(4x^2\)[/tex] from each term in the expression.
So,
[tex]\[ 4x^{10} - 64x^2 = 4x^2 (x^8 - 16) \][/tex]
### Part B: Factoring the Entire Expression Completely
Now, we look to further factor the expression inside the parentheses: [tex]\(x^8 - 16\)[/tex].
3. Recognize the Form of Difference of Squares:
- The expression [tex]\(x^8 - 16\)[/tex] is a difference of squares because [tex]\(x^8\)[/tex] and [tex]\(16\)[/tex] are both perfect squares.
- Recall that [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
Here,
[tex]\[ x^8 - 16 = (x^4)^2 - 4^2 \][/tex]
Using the difference of squares formula, we factor it as:
[tex]\[ (x^8 - 16) = (x^4 - 4)(x^4 + 4) \][/tex]
4. Further Factor [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]
5. Combine All the Factors:
- Now we have factored every part completely.
Therefore, the complete factorization of the entire expression is:
[tex]\[ 4x^2 (x^8 - 16) = 4x^2 (x^4 - 4)(x^4 + 4) = 4x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
So, the step-by-step solution is:
Part A: [tex]\(4x^{10} - 64x^2 = 4x^2 (x^8 - 16)\)[/tex]
Part B: Factoring completely:
[tex]\[ 4x^2 (x^8 - 16) = 4x^2 (x^4 - 4)(x^4 + 4) = 4x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
