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Answer :
To factor the given expression [tex]\(6x^6 + 2x^4 + 10x^3\)[/tex] by using the greatest common factor and the distributive property, let's go through the problem step-by-step:
1. Identify the Greatest Common Factor (GCF):
- Look at each term in the expression: [tex]\(6x^6\)[/tex], [tex]\(2x^4\)[/tex], and [tex]\(10x^3\)[/tex].
- The coefficients are 6, 2, and 10. The GCF of 6, 2, and 10 is 2.
- Next, consider the variable part. The lowest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^3\)[/tex].
- Therefore, the GCF of the whole expression is [tex]\(2x^3\)[/tex].
2. Factor Out the GCF:
- Divide each term by the GCF, [tex]\(2x^3\)[/tex]:
- [tex]\(6x^6 \div 2x^3 = 3x^3\)[/tex]
- [tex]\(2x^4 \div 2x^3 = x\)[/tex]
- [tex]\(10x^3 \div 2x^3 = 5\)[/tex]
- Write the expression in factored form using the GCF and the results from above:
[tex]\[
6x^6 + 2x^4 + 10x^3 = 2x^3(3x^3 + x + 5)
\][/tex]
This factored expression is equivalent to the original expression, using the greatest common factor combined with the distributive property.
1. Identify the Greatest Common Factor (GCF):
- Look at each term in the expression: [tex]\(6x^6\)[/tex], [tex]\(2x^4\)[/tex], and [tex]\(10x^3\)[/tex].
- The coefficients are 6, 2, and 10. The GCF of 6, 2, and 10 is 2.
- Next, consider the variable part. The lowest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^3\)[/tex].
- Therefore, the GCF of the whole expression is [tex]\(2x^3\)[/tex].
2. Factor Out the GCF:
- Divide each term by the GCF, [tex]\(2x^3\)[/tex]:
- [tex]\(6x^6 \div 2x^3 = 3x^3\)[/tex]
- [tex]\(2x^4 \div 2x^3 = x\)[/tex]
- [tex]\(10x^3 \div 2x^3 = 5\)[/tex]
- Write the expression in factored form using the GCF and the results from above:
[tex]\[
6x^6 + 2x^4 + 10x^3 = 2x^3(3x^3 + x + 5)
\][/tex]
This factored expression is equivalent to the original expression, using the greatest common factor combined with the distributive property.
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