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Sagot :
Certainly! Let's go through the steps to solve the given expression \(4x^{10} - 64x^2\).
### Part A: Factoring Out the Greatest Common Factor (GCF)
To begin, we need to determine the greatest common factor (GCF) of the terms \(4x^{10}\) and \(64x^2\).
1. Identify the GCF of the coefficients:
- The coefficients are 4 and 64.
- The GCF of 4 and 64 is 4.
2. Identify the GCF of the variable terms:
- The variable terms are \(x^{10}\) and \(x^2\).
- The GCF of \(x^{10}\) and \(x^2\) is \(x^2\).
Therefore, the GCF of the entire expression is \(4x^2\).
3. Factor out the GCF from each term:
- When you factor \(4x^2\) out of \(4x^{10}\), you are left with \(x^8\).
- When you factor \(4x^2\) out of \(64x^2\), you are left with 16.
Putting this together, we rewrite the expression as:
[tex]\[ 4x^{10} - 64x^2 = 4x^2 (x^8 - 16) \][/tex]
Thus, for Part A, the expression \(4x^{10} - 64x^2\) when factored out by the GCF is:
[tex]\[ 4x^2 (x^8 - 16) \][/tex]
### Part B: Factoring the Entire Expression Completely
Now, we need to factor \(x^8 - 16\) completely using the appropriate factoring techniques. The expression \(x^8 - 16\) is a difference of squares.
1. Recognize that \(x^8 - 16\) is a difference of squares:
[tex]\[ x^8 - 16 = (x^4)^2 - (4)^2 \][/tex]
- This can be written as:
[tex]\[ (x^4 - 4)(x^4 + 4) \][/tex]
2. Factor \(x^4 - 4\) further using the difference of squares formula:
- Notice that \(x^4 - 4\) is again a difference of squares:
[tex]\[ x^4 - 4 = (x^2)^2 - (2)^2 \][/tex]
- This can be factored as:
[tex]\[ (x^2 - 2)(x^2 + 2) \][/tex]
Putting everything together, we have:
[tex]\[ x^8 - 16 = (x^4 - 4)(x^4 + 4) = (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
Thus, the entire expression \(4x^2 (x^8 - 16)\) completely factored is:
[tex]\[ 4x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
### Final Answer:
- Part A: The expression \(4x^{10} - 64x^2\) factors with the GCF as \(4x^2 (x^8 - 16)\).
- Part B: The completely factored form of the expression is [tex]\(4x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4)\)[/tex].
### Part A: Factoring Out the Greatest Common Factor (GCF)
To begin, we need to determine the greatest common factor (GCF) of the terms \(4x^{10}\) and \(64x^2\).
1. Identify the GCF of the coefficients:
- The coefficients are 4 and 64.
- The GCF of 4 and 64 is 4.
2. Identify the GCF of the variable terms:
- The variable terms are \(x^{10}\) and \(x^2\).
- The GCF of \(x^{10}\) and \(x^2\) is \(x^2\).
Therefore, the GCF of the entire expression is \(4x^2\).
3. Factor out the GCF from each term:
- When you factor \(4x^2\) out of \(4x^{10}\), you are left with \(x^8\).
- When you factor \(4x^2\) out of \(64x^2\), you are left with 16.
Putting this together, we rewrite the expression as:
[tex]\[ 4x^{10} - 64x^2 = 4x^2 (x^8 - 16) \][/tex]
Thus, for Part A, the expression \(4x^{10} - 64x^2\) when factored out by the GCF is:
[tex]\[ 4x^2 (x^8 - 16) \][/tex]
### Part B: Factoring the Entire Expression Completely
Now, we need to factor \(x^8 - 16\) completely using the appropriate factoring techniques. The expression \(x^8 - 16\) is a difference of squares.
1. Recognize that \(x^8 - 16\) is a difference of squares:
[tex]\[ x^8 - 16 = (x^4)^2 - (4)^2 \][/tex]
- This can be written as:
[tex]\[ (x^4 - 4)(x^4 + 4) \][/tex]
2. Factor \(x^4 - 4\) further using the difference of squares formula:
- Notice that \(x^4 - 4\) is again a difference of squares:
[tex]\[ x^4 - 4 = (x^2)^2 - (2)^2 \][/tex]
- This can be factored as:
[tex]\[ (x^2 - 2)(x^2 + 2) \][/tex]
Putting everything together, we have:
[tex]\[ x^8 - 16 = (x^4 - 4)(x^4 + 4) = (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
Thus, the entire expression \(4x^2 (x^8 - 16)\) completely factored is:
[tex]\[ 4x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4) \][/tex]
### Final Answer:
- Part A: The expression \(4x^{10} - 64x^2\) factors with the GCF as \(4x^2 (x^8 - 16)\).
- Part B: The completely factored form of the expression is [tex]\(4x^2 (x^2 - 2)(x^2 + 2)(x^4 + 4)\)[/tex].
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