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Answer :
To factor out the greatest common factor (GCF) from the expression [tex]\(10x^6 + 25x^4 + 6x^3\)[/tex], let's go through the process step-by-step:
1. Identify the GCF of the coefficients:
The coefficients of the terms are 10, 25, and 6. We need to find the greatest common factor of these numbers.
- The factors of 10 are: 1, 2, 5, 10
- The factors of 25 are: 1, 5, 25
- The factors of 6 are: 1, 2, 3, 6
The greatest common factor that appears in all three lists is 1.
2. Identify the GCF of the variable parts:
The variable parts are [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex]. Here, you take the smallest power of [tex]\(x\)[/tex] amongst these terms, which is [tex]\(x^3\)[/tex].
3. Factor out the GCF:
The GCF of the entire expression is therefore [tex]\(1x^3\)[/tex] or simply [tex]\(x^3\)[/tex]. Now, factor [tex]\(x^3\)[/tex] out from each term:
- [tex]\(10x^6\)[/tex] becomes [tex]\((10x^3)(x^3)\)[/tex]
- [tex]\(25x^4\)[/tex] becomes [tex]\((25x^3)(x^1)\)[/tex]
- [tex]\(6x^3\)[/tex] becomes [tex]\((6x^3)(1)\)[/tex]
4. Write the factored expression:
Once you factor out [tex]\(x^3\)[/tex], the expression becomes:
[tex]\[
x^3(10x^3 + 25x + 6)
\][/tex]
So, the expression [tex]\(10x^6 + 25x^4 + 6x^3\)[/tex] factored by the GCF is [tex]\(x^3(10x^3 + 25x + 6)\)[/tex].
1. Identify the GCF of the coefficients:
The coefficients of the terms are 10, 25, and 6. We need to find the greatest common factor of these numbers.
- The factors of 10 are: 1, 2, 5, 10
- The factors of 25 are: 1, 5, 25
- The factors of 6 are: 1, 2, 3, 6
The greatest common factor that appears in all three lists is 1.
2. Identify the GCF of the variable parts:
The variable parts are [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex]. Here, you take the smallest power of [tex]\(x\)[/tex] amongst these terms, which is [tex]\(x^3\)[/tex].
3. Factor out the GCF:
The GCF of the entire expression is therefore [tex]\(1x^3\)[/tex] or simply [tex]\(x^3\)[/tex]. Now, factor [tex]\(x^3\)[/tex] out from each term:
- [tex]\(10x^6\)[/tex] becomes [tex]\((10x^3)(x^3)\)[/tex]
- [tex]\(25x^4\)[/tex] becomes [tex]\((25x^3)(x^1)\)[/tex]
- [tex]\(6x^3\)[/tex] becomes [tex]\((6x^3)(1)\)[/tex]
4. Write the factored expression:
Once you factor out [tex]\(x^3\)[/tex], the expression becomes:
[tex]\[
x^3(10x^3 + 25x + 6)
\][/tex]
So, the expression [tex]\(10x^6 + 25x^4 + 6x^3\)[/tex] factored by the GCF is [tex]\(x^3(10x^3 + 25x + 6)\)[/tex].
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