Let's factor the expression [tex]\(3y^2 + 15y + 18\)[/tex] completely.
Step 1: Identify the common factor.
We start by checking if there is a common factor for all the terms in the expression.
The terms are:
- [tex]\(3y^2\)[/tex]
- [tex]\(15y\)[/tex]
- [tex]\(18\)[/tex]
The greatest common factor (GCF) for the coefficients 3, 15, and 18 is 3. So, we can factor out a 3 from the entire expression:
[tex]\[3y^2 + 15y + 18 = 3(y^2 + 5y + 6)\][/tex]
Step 2: Factor the quadratic expression inside the parentheses.
Now, we need to factor the quadratic expression [tex]\(y^2 + 5y + 6\)[/tex].
We look for two numbers that multiply to the constant term 6 and add up to the coefficient of the middle term, 5.
The numbers 2 and 3 work because:
[tex]\[
2 \cdot 3 = 6 \quad \text{and} \quad 2 + 3 = 5
\][/tex]
Therefore, we can factor the quadratic expression as:
[tex]\[ y^2 + 5y + 6 = (y + 2)(y + 3) \][/tex]
Step 3: Combine the factored forms.
Substitute the factored quadratic expression back:
[tex]\[ 3(y^2 + 5y + 6) = 3(y + 2)(y + 3) \][/tex]
Conclusion:
The expression [tex]\(3y^2 + 15y + 18\)[/tex] factors completely as:
[tex]\[ 3(y + 2)(y + 3) \][/tex]
This is the fully factored form of the given expression.
