Sure, let's break down the expression [tex]\( 12a^3 + 10a^2 \)[/tex] step-by-step to simplify it.
### Step 1: Identify the Greatest Common Factor (GCF)
Firstly, we need to identify the greatest common factor (GCF) of the terms in the expression.
- For the numerical coefficients (12 and 10):
- The factors of 12 are: [tex]\(1, 2, 3, 4, 6, 12\)[/tex]
- The factors of 10 are: [tex]\(1, 2, 5, 10\)[/tex]
- The GCF of 12 and 10 is: [tex]\(2\)[/tex]
- For the variable part ([tex]\(a^3\)[/tex] and [tex]\(a^2\)[/tex]):
- The minimum power of [tex]\(a\)[/tex] common to both terms is [tex]\(a^2\)[/tex]
Therefore, the GCF of the entire expression is [tex]\(2a^2\)[/tex].
### Step 2: Factor Out the GCF
We now factor out [tex]\(2a^2\)[/tex] from each term in the expression [tex]\(12a^3 + 10a^2\)[/tex]:
1. Divide [tex]\(12a^3\)[/tex] by [tex]\(2a^2\)[/tex]:
[tex]\[
\frac{12a^3}{2a^2} = 6a
\][/tex]
2. Divide [tex]\(10a^2\)[/tex] by [tex]\(2a^2\)[/tex]:
[tex]\[
\frac{10a^2}{2a^2} = 5
\][/tex]
### Step 3: Write the Factored Expression
After factoring out [tex]\(2a^2\)[/tex] from each term, we can write the expression as:
[tex]\[
12a^3 + 10a^2 = 2a^2 (6a + 5)
\][/tex]
### Final Answer:
Thus, the factored form of the expression [tex]\(12a^3 + 10a^2\)[/tex] is:
[tex]\[
2a^2 (6a + 5)
\][/tex]
