Answer :
Sure! Let's solve the problem step by step.
We need to write the expression [tex]\(12x^5 + 20x^3 + 4x^2\)[/tex] as a product by factoring out the greatest common factor (GCF).
### Step 1: Identify the GCF of the coefficients
The coefficients of the terms are 12, 20, and 4. To find their greatest common factor, we need to determine the largest number that evenly divides all three coefficients.
- The factors of 12 are: 1, 2, 3, 4, 6, 12
- The factors of 20 are: 1, 2, 4, 5, 10, 20
- The factors of 4 are: 1, 2, 4
The common factors are 1, 2, and 4. The greatest of these is 4.
So, the greatest common factor of the coefficients is 4.
### Step 2: Identify the GCF of the variable parts
We have [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]. The GCF for [tex]\(x\)[/tex]-terms is the lowest power of [tex]\(x\)[/tex] that appears in each term, which is [tex]\(x^2\)[/tex].
### Step 3: Factor out the GCF from each term
Now, we'll factor out [tex]\( 4x^2 \)[/tex] from each term in the expression:
[tex]\[
12x^5 + 20x^3 + 4x^2 = 4x^2 \cdot \left(\frac{12x^5}{4x^2} + \frac{20x^3}{4x^2} + \frac{4x^2}{4x^2} \right)
\][/tex]
Simplify each fraction inside the parentheses:
[tex]\[
4x^2 \cdot \left( 3x^3 + 5x + 1 \right)
\][/tex]
So, when we factor out the greatest common factor, we get:
[tex]\[
12x^5 + 20x^3 + 4x^2 = 4x^2 \left( 3x^3 + 5x + 1 \right)
\][/tex]
### Final Answer
The expression [tex]\(\boldsymbol{12x^5 + 20x^3 + 4x^2}\)[/tex] can be written as a product of its greatest common factor:
[tex]\(\boldsymbol{4x^2 \left( 3x^3 + 5x + 1 \right)}\)[/tex].
We need to write the expression [tex]\(12x^5 + 20x^3 + 4x^2\)[/tex] as a product by factoring out the greatest common factor (GCF).
### Step 1: Identify the GCF of the coefficients
The coefficients of the terms are 12, 20, and 4. To find their greatest common factor, we need to determine the largest number that evenly divides all three coefficients.
- The factors of 12 are: 1, 2, 3, 4, 6, 12
- The factors of 20 are: 1, 2, 4, 5, 10, 20
- The factors of 4 are: 1, 2, 4
The common factors are 1, 2, and 4. The greatest of these is 4.
So, the greatest common factor of the coefficients is 4.
### Step 2: Identify the GCF of the variable parts
We have [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]. The GCF for [tex]\(x\)[/tex]-terms is the lowest power of [tex]\(x\)[/tex] that appears in each term, which is [tex]\(x^2\)[/tex].
### Step 3: Factor out the GCF from each term
Now, we'll factor out [tex]\( 4x^2 \)[/tex] from each term in the expression:
[tex]\[
12x^5 + 20x^3 + 4x^2 = 4x^2 \cdot \left(\frac{12x^5}{4x^2} + \frac{20x^3}{4x^2} + \frac{4x^2}{4x^2} \right)
\][/tex]
Simplify each fraction inside the parentheses:
[tex]\[
4x^2 \cdot \left( 3x^3 + 5x + 1 \right)
\][/tex]
So, when we factor out the greatest common factor, we get:
[tex]\[
12x^5 + 20x^3 + 4x^2 = 4x^2 \left( 3x^3 + 5x + 1 \right)
\][/tex]
### Final Answer
The expression [tex]\(\boldsymbol{12x^5 + 20x^3 + 4x^2}\)[/tex] can be written as a product of its greatest common factor:
[tex]\(\boldsymbol{4x^2 \left( 3x^3 + 5x + 1 \right)}\)[/tex].
