To factor the quadratic expression [tex]\(3y^2 + 78y + 264\)[/tex] completely, follow these steps:
1. Identify the coefficients:
The quadratic expression is in the form [tex]\(ay^2 + by + c\)[/tex], where:
[tex]\[
a = 3, \quad b = 78, \quad c = 264
\][/tex]
2. Check for a common factor:
First, see if there is a common factor in all terms of the expression.
[tex]\[
\text{The greatest common factor (GCF) of } 3, 78, \text{ and } 264 \text{ is } 3.
\][/tex]
Factor out the GCF:
[tex]\[
3y^2 + 78y + 264 = 3(y^2 + 26y + 88)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
Now, we need to factorize the quadratic expression [tex]\(y^2 + 26y + 88\)[/tex].
4. Find two numbers that multiply to the constant term (88) and add to the linear coefficient (26):
Consider the factors of 88. We need two numbers that multiply to 88 and sum to 26. The pairs of factors are:
[tex]\[
1 \times 88, \; 2 \times 44, \; 4 \times 22, \; 8 \times 11
\][/tex]
The correct pair is [tex]\(4\)[/tex] and [tex]\(22\)[/tex].
5. Rewrite the quadratic expression using these two numbers:
We can split the middle term, [tex]\(26y\)[/tex], using the two numbers we found:
[tex]\[
y^2 + 26y + 88 = y^2 + 4y + 22y + 88
\][/tex]
6. Factor by grouping:
Group the terms in pairs and factor each pair:
[tex]\[
y^2 + 4y + 22y + 88 = y(y + 4) + 22(y + 4)
\][/tex]
7. Factor out the common binomial factor:
The expression can now be factored by taking out the common binomial factor [tex]\((y + 4)\)[/tex]:
[tex]\[
y(y + 4) + 22(y + 4) = (y + 4)(y + 22)
\][/tex]
8. Combine with the factor from step 2:
Multiply back by the greatest common factor [tex]\(3\)[/tex] we factored out in step 2:
[tex]\[
3(y^2 + 26y + 88) = 3(y + 4)(y + 22)
\][/tex]
Hence, the completely factored form of the expression [tex]\(3y^2 + 78y + 264\)[/tex] is:
[tex]\[
3(y + 4)(y + 22)
\][/tex]
