Answer :
Sure! Let's go through the steps to factor the expression [tex]\( 25x^3 - 10x^2 \)[/tex] using the Greatest Common Factor (GCF).
### Step-by-Step Solution
1. Identify the GCF:
- First, we look for the greatest common factor in the coefficients of the terms.
- The coefficients are 25 and 10.
- The GCF of 25 and 10 is 5.
- Next, we look for the greatest common factor in the variable part.
- The terms are [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex].
- The GCF of [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex] is [tex]\( x^2 \)[/tex].
- Combining the two, the overall GCF of the expression is [tex]\( 5x^2 \)[/tex].
2. Factor out the GCF:
- We now factor the GCF, [tex]\( 5x^2 \)[/tex], out of each term of the expression.
- The original expression is [tex]\( 25x^3 - 10x^2 \)[/tex].
- Factoring [tex]\( 5x^2 \)[/tex] out of the first term:
[tex]\[
25x^3 = 5x^2 \cdot 5x
\][/tex]
- Factoring [tex]\( 5x^2 \)[/tex] out of the second term:
[tex]\[
10x^2 = 5x^2 \cdot 2
\][/tex]
3. Rewrite the expression:
- After factoring out the [tex]\( 5x^2 \)[/tex], we combine the factors from both terms and write the expression in a factored form:
[tex]\[
25x^3 - 10x^2 = 5x^2 (5x) - 5x^2 (2)
\][/tex]
- This simplifies to:
[tex]\[
25x^3 - 10x^2 = 5x^2 (5x - 2)
\][/tex]
### Final Answer
The factored form of the expression [tex]\( 25x^3 - 10x^2 \)[/tex] is:
[tex]\[
5x^2 (5x - 2)
\][/tex]
This is our final result after factoring using the Greatest Common Factor (GCF).
### Step-by-Step Solution
1. Identify the GCF:
- First, we look for the greatest common factor in the coefficients of the terms.
- The coefficients are 25 and 10.
- The GCF of 25 and 10 is 5.
- Next, we look for the greatest common factor in the variable part.
- The terms are [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex].
- The GCF of [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex] is [tex]\( x^2 \)[/tex].
- Combining the two, the overall GCF of the expression is [tex]\( 5x^2 \)[/tex].
2. Factor out the GCF:
- We now factor the GCF, [tex]\( 5x^2 \)[/tex], out of each term of the expression.
- The original expression is [tex]\( 25x^3 - 10x^2 \)[/tex].
- Factoring [tex]\( 5x^2 \)[/tex] out of the first term:
[tex]\[
25x^3 = 5x^2 \cdot 5x
\][/tex]
- Factoring [tex]\( 5x^2 \)[/tex] out of the second term:
[tex]\[
10x^2 = 5x^2 \cdot 2
\][/tex]
3. Rewrite the expression:
- After factoring out the [tex]\( 5x^2 \)[/tex], we combine the factors from both terms and write the expression in a factored form:
[tex]\[
25x^3 - 10x^2 = 5x^2 (5x) - 5x^2 (2)
\][/tex]
- This simplifies to:
[tex]\[
25x^3 - 10x^2 = 5x^2 (5x - 2)
\][/tex]
### Final Answer
The factored form of the expression [tex]\( 25x^3 - 10x^2 \)[/tex] is:
[tex]\[
5x^2 (5x - 2)
\][/tex]
This is our final result after factoring using the Greatest Common Factor (GCF).
