Sure, let's tackle the problem step by step.
Given Expression:
[tex]\[ 4x^{10} - 64x^2 \][/tex]
### Part A: Factoring out the Greatest Common Factor (GCF)
1. Identify the terms:
- First term: [tex]\( 4x^{10} \)[/tex]
- Second term: [tex]\( 64x^2 \)[/tex]
2. Determine the greatest common factor (GCF):
- The GCF of the coefficients 4 and 64 is 4.
- The GCF of the variables [tex]\(x^{10}\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex] (since [tex]\(x^2\)[/tex] is the highest power of [tex]\(x\)[/tex] that is common to both terms).
Therefore, the GCF of the entire expression is [tex]\(4x^2\)[/tex].
3. Factor out the GCF:
- [tex]\( 4x^{10} - 64x^2 = 4x^2 \cdot (x^8) - 4x^2 \cdot (16) \)[/tex]
- Factoring out [tex]\(4x^2\)[/tex] gives:
[tex]\[ 4x^{10} - 64x^2 = 4x^2 (x^8 - 16) \][/tex]
So, the expression factored by the GCF is:
[tex]\[ 4x^2 (x^8 - 16) \][/tex]
### Part B: Factor the Entire Expression Completely
1. Factor the expression inside the parenthesis:
- The expression [tex]\( x^8 - 16 \)[/tex] can be seen as a difference of squares.
- Recall that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
2. Recognize the difference of squares:
- [tex]\( x^8 \)[/tex] is a perfect square: [tex]\( (x^4)^2 \)[/tex].
- 16 is a perfect square: [tex]\( 4^2 \)[/tex].
Therefore:
[tex]\[ x^8 - 16 = (x^4)^2 - 4^2 = (x^4 - 4)(x^4 + 4) \][/tex]
3. Further factor each part, if possible:
- For [tex]\( x^4 - 4 \)[/tex]:
[tex]\[ x^4 - 4 = (x^2)^2 - 2^2 = (x^2 - 2)(x^2 + 2) \][/tex]
- For [tex]\( x^4 + 4 \)[/tex]:
This can be tricky, but it can be factored further into:
[tex]\[ x^4 + 4 = (x^2 - \sqrt{2}x + 2)(x^2 + \sqrt{2}x + 2) \][/tex]
However, it is conventional to use integer coefficients when possible, so we can use another commonly taught factorization:
[tex]\[ x^4 + 4 = (x^2 - 2x + 2)(x^2 + 2x + 2) \][/tex]
Combining all parts, the completely factored form of the expression is:
[tex]\[ 4x^2 \left( (x^2 - 2)(x^2 + 2)(x^2 - 2x + 2)(x^2 + 2x + 2) \right) \][/tex]
### Summary
- Part A: The expression factored by the GCF is:
[tex]\[ 4x^2 (x^8 - 16) \][/tex]
- Part B: The completely factored form of the expression is:
[tex]\[ 4x^2 (x^2 - 2)(x^2 + 2)(x^2 - 2x + 2)(x^2 + 2x + 2) \][/tex]
