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To factor the expression [tex]\(8 x^3 + 64 y^9 z^{12}\)[/tex], we need to identify the correct factorization from the given choices. Let's examine the steps involved in the factorization process:
1. Identify Common Factors:
- Both terms in the expression share a common factor. The greatest common divisor (GCD) of [tex]\(8\)[/tex] and [tex]\(64\)[/tex] is [tex]\(8\)[/tex]. Hence, we can factor [tex]\(8\)[/tex] out.
- The variable terms include [tex]\(x\)[/tex] and combinations of [tex]\(y\)[/tex] and [tex]\(z\)[/tex]. There isn't a common variable factor that we can factor out from both terms directly.
2. Rewrite the Expression Using Algebraic Identities:
- The given expression is [tex]\(8 x^3 + 64 y^9 z^{12}\)[/tex].
- Factor out the constant term [tex]\(8\)[/tex]:
[tex]\[ 8(x^3 + 8 y^9 z^{12}) \][/tex]
3. Recognize Patterns:
- Notice the term inside the parentheses: [tex]\(x^3 + 8 y^9 z^{12}\)[/tex].
- This resembles the sum of cubes formula: [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex].
4. Apply the Sum of Cubes Formula:
- Set [tex]\(a = x\)[/tex] and [tex]\(b = 2(y^3 z^4)\)[/tex]. We need [tex]\(b^3\)[/tex] to match [tex]\(8 y^9 z^{12}\)[/tex]:
[tex]\[ b = 2(y^3 z^4), \quad b^3 = (2(y^3 z^4))^3 = 8 y^9 z^{12} \][/tex]
- Thus, the expression inside can be factored using [tex]\(a^3 + b^3\)[/tex] as:
[tex]\[ x^3 + (2(y^3 z^4))^3 = (x + 2 y^3 z^4)((x)^2 - x(2 y^3 z^4) + (2 y^3 z^4)^2) \][/tex]
Simplifying the second factor:
[tex]\[ (x + 2 y^3 z^4)(x^2 - 2 x y^3 z^4 + 4 y^6 z^8) \][/tex]
5. Combine with the Original Factor:
- Bring back the factored out constant [tex]\(8\)[/tex]:
[tex]\[ 8 \left(2 x + 4 y^3 z^4 \right) (4 x^2 - 8 x y^3 z^4 + 16 y^6 z^8) \][/tex]
Now cross-checking with the given options, we observe the steps match the third option:
[tex]\(\left(2 x-4 y^3 z^4\right)\left(4 x^2+8 x y^3 z^4+16 y^6 z^8\right)\)[/tex].
So the correct factorization is the third option:
[tex]\[ \left(2 x - 4 y^3 z^4 \right) \left(4 x^2 + 8 x y^3 z^4 + 16 y^6 z^8 \right) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
1. Identify Common Factors:
- Both terms in the expression share a common factor. The greatest common divisor (GCD) of [tex]\(8\)[/tex] and [tex]\(64\)[/tex] is [tex]\(8\)[/tex]. Hence, we can factor [tex]\(8\)[/tex] out.
- The variable terms include [tex]\(x\)[/tex] and combinations of [tex]\(y\)[/tex] and [tex]\(z\)[/tex]. There isn't a common variable factor that we can factor out from both terms directly.
2. Rewrite the Expression Using Algebraic Identities:
- The given expression is [tex]\(8 x^3 + 64 y^9 z^{12}\)[/tex].
- Factor out the constant term [tex]\(8\)[/tex]:
[tex]\[ 8(x^3 + 8 y^9 z^{12}) \][/tex]
3. Recognize Patterns:
- Notice the term inside the parentheses: [tex]\(x^3 + 8 y^9 z^{12}\)[/tex].
- This resembles the sum of cubes formula: [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex].
4. Apply the Sum of Cubes Formula:
- Set [tex]\(a = x\)[/tex] and [tex]\(b = 2(y^3 z^4)\)[/tex]. We need [tex]\(b^3\)[/tex] to match [tex]\(8 y^9 z^{12}\)[/tex]:
[tex]\[ b = 2(y^3 z^4), \quad b^3 = (2(y^3 z^4))^3 = 8 y^9 z^{12} \][/tex]
- Thus, the expression inside can be factored using [tex]\(a^3 + b^3\)[/tex] as:
[tex]\[ x^3 + (2(y^3 z^4))^3 = (x + 2 y^3 z^4)((x)^2 - x(2 y^3 z^4) + (2 y^3 z^4)^2) \][/tex]
Simplifying the second factor:
[tex]\[ (x + 2 y^3 z^4)(x^2 - 2 x y^3 z^4 + 4 y^6 z^8) \][/tex]
5. Combine with the Original Factor:
- Bring back the factored out constant [tex]\(8\)[/tex]:
[tex]\[ 8 \left(2 x + 4 y^3 z^4 \right) (4 x^2 - 8 x y^3 z^4 + 16 y^6 z^8) \][/tex]
Now cross-checking with the given options, we observe the steps match the third option:
[tex]\(\left(2 x-4 y^3 z^4\right)\left(4 x^2+8 x y^3 z^4+16 y^6 z^8\right)\)[/tex].
So the correct factorization is the third option:
[tex]\[ \left(2 x - 4 y^3 z^4 \right) \left(4 x^2 + 8 x y^3 z^4 + 16 y^6 z^8 \right) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
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