Let's factor the polynomial [tex]\(12y^4 - 42y^3 + 18y^2\)[/tex] step-by-step.
1. Identify common factors in the polynomial:
The first step is to look for a greatest common factor (GCF) among the terms in the polynomial. Here, we notice that each term in [tex]\(12y^4 - 42y^3 + 18y^2\)[/tex] has a common factor of [tex]\(6y^2\)[/tex]. Thus, we can factor out [tex]\(6y^2\)[/tex] from each term:
[tex]\[
12y^4 - 42y^3 + 18y^2 = 6y^2(2y^2) - 6y^2(7y) + 6y^2(3)
\][/tex]
Simplifying this gives us:
[tex]\[
12y^4 - 42y^3 + 18y^2 = 6y^2(2y^2 - 7y + 3)
\][/tex]
Now we need to focus on factoring the quadratic expression [tex]\(2y^2 - 7y + 3\)[/tex].
2. Factor the quadratic expression:
We need to factor [tex]\(2y^2 - 7y + 3\)[/tex]. To factor it, we look for two binomials of the form [tex]\((ay + b)(cy + d)\)[/tex] such that:
[tex]\[
(ay + b)(cy + d) = acy^2 + (ad + bc)y + bd
\][/tex]
For our quadratic, [tex]\(ac = 2 \cdot 1 = 2\)[/tex] and [tex]\(bd = 1 \cdot 3 = 3\)[/tex], and the middle term coefficient [tex]\(-7\)[/tex] should be the sum [tex]\(ad + bc\)[/tex]. Thus, we need:
[tex]\[
2y^2 - 7y + 3
\][/tex]
Here is the factorization breakdown:
- The coefficient of [tex]\(y^2\)[/tex] is 2, which we keep in mind.
- The constant term is 3.
- Find two numbers that multiply to [tex]\(2 \times 3 = 6\)[/tex] and add to [tex]\(-7\)[/tex]. These numbers are [tex]\(-1\)[/tex] and [tex]\(-6\)[/tex].
Let's rewrite the middle term using these numbers:
[tex]\[
2y^2 - y - 6y + 3
\][/tex]
Group the terms:
[tex]\[
(2y^2 - y) + (-6y + 3)
\][/tex]
Factor out the common factors from each group:
[tex]\[
y(2y - 1) - 3(2y - 1)
\][/tex]
Notice that [tex]\( (2y - 1) \)[/tex] is common in both terms:
[tex]\[
(2y - 1)(y - 3)
\][/tex]
3. Combine the factors:
Now, combine this factored form back with the common factor we originally factored out:
[tex]\[
12y^4 - 42y^3 + 18y^2 = 6y^2(2y - 1)(y - 3)
\][/tex]
So, the fully factored form of the polynomial [tex]\(12y^4 - 42y^3 + 18y^2\)[/tex] is:
[tex]\[
6y^2 (2y - 1)(y - 3)
\][/tex]
This is the detailed step-by-step solution for factoring the polynomial.
