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Answer :
To factor the expression [tex]\(10x^7 + 35x^6 + 25x^5\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
Look at the coefficients and the exponents of [tex]\(x\)[/tex] in each term. The coefficients are 10, 35, and 25, which share a GCF of 5. For the variables, the smallest power of [tex]\(x\)[/tex] in each term is [tex]\(x^5\)[/tex]. Therefore, the GCF for the entire expression is [tex]\(5x^5\)[/tex].
2. Factor Out the GCF:
Divide each term of the expression by the GCF, which is [tex]\(5x^5\)[/tex]:
[tex]\[
\frac{10x^7}{5x^5} = 2x^2
\][/tex]
[tex]\[
\frac{35x^6}{5x^5} = 7x
\][/tex]
[tex]\[
\frac{25x^5}{5x^5} = 5
\][/tex]
This results in the expression becoming:
[tex]\[
5x^5(2x^2 + 7x + 5)
\][/tex]
3. Verify if the Quadratic Can Be Factored Further:
The expression inside the parentheses, [tex]\(2x^2 + 7x + 5\)[/tex], cannot be factored further using integers, as it doesn't have factors that simplify it more.
Thus, the completely factored form of the given expression is:
[tex]\[
5x^5(2x^2 + 7x + 5)
\][/tex]
This is the final answer.
1. Identify the Greatest Common Factor (GCF):
Look at the coefficients and the exponents of [tex]\(x\)[/tex] in each term. The coefficients are 10, 35, and 25, which share a GCF of 5. For the variables, the smallest power of [tex]\(x\)[/tex] in each term is [tex]\(x^5\)[/tex]. Therefore, the GCF for the entire expression is [tex]\(5x^5\)[/tex].
2. Factor Out the GCF:
Divide each term of the expression by the GCF, which is [tex]\(5x^5\)[/tex]:
[tex]\[
\frac{10x^7}{5x^5} = 2x^2
\][/tex]
[tex]\[
\frac{35x^6}{5x^5} = 7x
\][/tex]
[tex]\[
\frac{25x^5}{5x^5} = 5
\][/tex]
This results in the expression becoming:
[tex]\[
5x^5(2x^2 + 7x + 5)
\][/tex]
3. Verify if the Quadratic Can Be Factored Further:
The expression inside the parentheses, [tex]\(2x^2 + 7x + 5\)[/tex], cannot be factored further using integers, as it doesn't have factors that simplify it more.
Thus, the completely factored form of the given expression is:
[tex]\[
5x^5(2x^2 + 7x + 5)
\][/tex]
This is the final answer.
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