To factor the expression [tex]\(3y^2 + 15y + 18\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, check if there is a common factor in all the terms of the expression. In this case, each term in [tex]\(3y^2 + 15y + 18\)[/tex] is divisible by 3.
So, we can factor out the 3:
[tex]\[
3y^2 + 15y + 18 = 3(y^2 + 5y + 6)
\][/tex]
2. Factor the Quadratic Expression:
Next, we need to factor the quadratic expression [tex]\(y^2 + 5y + 6\)[/tex]. To do this, we look for two numbers that multiply to the constant term (6) and add up to the linear coefficient (5).
The pairs of numbers that multiply to 6 are:
- (1, 6)
- (2, 3)
Among these pairs, the pair that adds up to 5 is (2, 3). Therefore, we can write:
[tex]\[
y^2 + 5y + 6 = (y + 2)(y + 3)
\][/tex]
3. Combine the Factored Terms:
Finally, substitute the factored quadratic back into the expression we factored out in step 1:
[tex]\[
3(y^2 + 5y + 6) = 3(y + 2)(y + 3)
\][/tex]
Thus, the completely factored form of [tex]\(3y^2 + 15y + 18\)[/tex] is:
[tex]\[
3(y + 2)(y + 3)
\][/tex]
