Determining Factors by Grouping
To find the factors of the terms 12x³, 2x², and 18x³, we can use the method of grouping. This method involves identifying common factors in pairs of terms.
Step 1: Write down the terms
We start with:
- 12x³
- 2x²
- 18x³
Step 2: Group the terms
We can group the terms in pairs:
- (12x³ and 18x³)
- (2x²)
Step 3: Factor out the greatest common factor (GCF)
Let’s first find the GCF for each of the groups:
- For 12x³ and 18x³:
- The numerical GCF of 12 and 18 is 6.
- The variable GCF is x³ (since both have x³).
So, the GCF is 6x³.
- For 2x²:
Since it’s a single term, its GCF is itself 2x².
Step 4: Rewrite the expression
Now, we can rewrite the expression by factoring out the common factors:
We take the GCF of the first pair:
12x³ + 18x³ = 6x³(2 + 3) = 6x³(5)
So this gives us:
6x³ * 5 + 2x²
Step 5: Factor out overall GCF
Now let’s find the overall GCF from all terms. The GCF from all terms is:
- GCF of the numerical coefficients (12, 2, and 18) is 2.
- The variable part, from x³ and x², is x² (the lowest power).
Therefore, the overall GCF is 2x².
Final Factored Form
Now we can express the original terms as:
2x²(6x + 1)
This gives us the factored form by grouping.
Conclusion
Using grouping, we factored the polynomial expression of 12x³, 2x², and 18x³ into 2x²(6x + 1).