To factor the trinomial [tex]\(6x^2 - xy - 2y^2\)[/tex] completely, we will follow a systematic approach for factoring quadratic expressions of the form [tex]\(ax^2 + bxy + cy^2\)[/tex]. Here are the steps:
1. Identify the coefficients: The given trinomial is [tex]\(6x^2 - xy - 2y^2\)[/tex]. We can identify the coefficients as:
- [tex]\(a = 6\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = -1\)[/tex] (coefficient of [tex]\(xy\)[/tex])
- [tex]\(c = -2\)[/tex] (coefficient of [tex]\(y^2\)[/tex])
2. Set up the factor pairs: We need to find two binomials of the form [tex]\((mx + ny)\)[/tex] and [tex]\((px + qy)\)[/tex] that multiply together to give us the original trinomial. The product of the first terms must equal [tex]\(a\)[/tex], and the product of the last terms must equal [tex]\(c\)[/tex].
3. Find the correct factor pair: To guess the correct factors, we look for numbers that multiply to [tex]\(a \cdot c = 6 \cdot (-2) = -12\)[/tex] and add up to [tex]\(b = -1\)[/tex]. These two numbers are 2 and -3 because [tex]\(2 \cdot (-3) = -6\)[/tex] and [tex]\(2 + (-3) = -1\)[/tex].
4. Split the middle term: We rewrite the middle term [tex]\(-xy\)[/tex] using our factor pair:
[tex]\[
6x^2 - xy - 2y^2 = 6x^2 + 2xy - 3xy - 2y^2
\][/tex]
5. Group and factor by grouping:
- Group the terms to factor by grouping:
[tex]\[
(6x^2 + 2xy) + (-3xy - 2y^2)
\][/tex]
- Factor out the greatest common factor (GCF) from each group:
[tex]\[
2x(3x + y) - y(3x + 2y)
\][/tex]
6. Factor out the common binomial:
- Notice that [tex]\((3x + y)\)[/tex] is a common factor:
[tex]\[
(2x - y)(3x + 2y)
\][/tex]
So, the trinomial [tex]\(6x^2 - xy - 2y^2\)[/tex] factored completely is:
[tex]\[
(2x + y)(3x - 2y)
\][/tex]
Thus, the factored form of the trinomial [tex]\(6x^2 - xy - 2y^2\)[/tex] is [tex]\((2x + y)(3x - 2y)\)[/tex].
