Certainly! Let's simplify and factor the given expression step by step:
Given expression:
[tex]\[ 12c^2d^2 - 24cd^2 + 6c^3d^2 \][/tex]
### Step 1: Factor out the Greatest Common Factor (GCF)
First, identify the greatest common factor for all terms in the expression.
- Each term has a common factor of [tex]\(6cd^2\)[/tex]:
- [tex]\(12c^2d^2\)[/tex] can be written as [tex]\(6cd^2 \cdot 2c\)[/tex]
- [tex]\(24cd^2\)[/tex] can be written as [tex]\(6cd^2 \cdot 4\)[/tex]
- [tex]\(6c^3d^2\)[/tex] can be written as [tex]\(6cd^2 \cdot c^2\)[/tex]
So, the GCF is [tex]\(6cd^2\)[/tex]. We factor out [tex]\(6cd^2\)[/tex]:
[tex]\[ 12c^2d^2 - 24cd^2 + 6c^3d^2 = 6cd^2(2c - 4 + c^2) \][/tex]
### Step 2: Simplify the Remaining Expression
Now we simplify the expression inside the parentheses:
[tex]\[ 2c - 4 + c^2 = c^2 + 2c - 4 \][/tex]
Thus, we have:
[tex]\[ 6cd^2(c^2 + 2c - 4) \][/tex]
### Step 3: Final Expression
The fully factored form of the given expression is:
[tex]\[ 12c^2d^2 - 24cd^2 + 6c^3d^2 = 6cd^2(c^2 + 2c - 4) \][/tex]
And that is the simplified and factored form of the given polynomial.
