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Answer :
To factor out the greatest common factor (GCF) from the expression [tex]\(10x^5 + 25x^3\)[/tex], follow these steps:
1. Identify the coefficients and their GCF:
The expression has coefficients 10 and 25. The GCF of 10 and 25 is 5, as 5 is the largest number that divides both 10 and 25 without leaving a remainder.
2. Identify the variable part and its GCF:
The expression includes the terms [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex]. Both terms include the variable [tex]\(x\)[/tex], and the GCF of [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex] is [tex]\(x^3\)[/tex]. This is because [tex]\(x^3\)[/tex] is the highest power of [tex]\(x\)[/tex] that can divide both [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex].
3. Combine the GCFs:
Combine the numerical GCF and the variable GCF to get the overall GCF of the entire expression, which is [tex]\(5x^3\)[/tex].
4. Factor out the GCF from the expression:
Divide each term in the expression [tex]\(10x^5 + 25x^3\)[/tex] by the GCF, [tex]\(5x^3\)[/tex].
- Dividing [tex]\(10x^5\)[/tex] by [tex]\(5x^3\)[/tex] gives:
[tex]\[
\frac{10x^5}{5x^3} = 2x^2
\][/tex]
- Dividing [tex]\(25x^3\)[/tex] by [tex]\(5x^3\)[/tex] gives:
[tex]\[
\frac{25x^3}{5x^3} = 5
\][/tex]
5. Write the factored expression:
After factoring out the GCF, the original expression is rewritten as the product of the GCF and the simplified terms:
[tex]\[
10x^5 + 25x^3 = 5x^3(2x^2 + 5)
\][/tex]
Thus, the factored form of the expression is [tex]\(5x^3(2x^2 + 5)\)[/tex].
1. Identify the coefficients and their GCF:
The expression has coefficients 10 and 25. The GCF of 10 and 25 is 5, as 5 is the largest number that divides both 10 and 25 without leaving a remainder.
2. Identify the variable part and its GCF:
The expression includes the terms [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex]. Both terms include the variable [tex]\(x\)[/tex], and the GCF of [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex] is [tex]\(x^3\)[/tex]. This is because [tex]\(x^3\)[/tex] is the highest power of [tex]\(x\)[/tex] that can divide both [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex].
3. Combine the GCFs:
Combine the numerical GCF and the variable GCF to get the overall GCF of the entire expression, which is [tex]\(5x^3\)[/tex].
4. Factor out the GCF from the expression:
Divide each term in the expression [tex]\(10x^5 + 25x^3\)[/tex] by the GCF, [tex]\(5x^3\)[/tex].
- Dividing [tex]\(10x^5\)[/tex] by [tex]\(5x^3\)[/tex] gives:
[tex]\[
\frac{10x^5}{5x^3} = 2x^2
\][/tex]
- Dividing [tex]\(25x^3\)[/tex] by [tex]\(5x^3\)[/tex] gives:
[tex]\[
\frac{25x^3}{5x^3} = 5
\][/tex]
5. Write the factored expression:
After factoring out the GCF, the original expression is rewritten as the product of the GCF and the simplified terms:
[tex]\[
10x^5 + 25x^3 = 5x^3(2x^2 + 5)
\][/tex]
Thus, the factored form of the expression is [tex]\(5x^3(2x^2 + 5)\)[/tex].
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