Factor each trinomial, if possible. See Examples 3 and 4. $12x^2-xy-y^2$

Verified step by step guidance

1. Identify the trinomial to factor: \(12x^{2} - xy - y^{2}\).
2. Look for two numbers that multiply to the product of the coefficient of \(x^{2}\) term and the constant term, which is \(12 \times (-1) = -12\), and add up to the coefficient of the middle term, which is \(-1\).
3. Find the pair of numbers that satisfy these conditions. In this case, the numbers are \(3\) and \(-4\) because \(3 \times (-4) = -12\) and \(3 + (-4) = -1\).
4. Rewrite the middle term \(-xy\) using the two numbers found: \(12x^{2} + 3xy - 4xy - y^{2}\).
5. Group the terms and factor each group: \((12x^{2} + 3xy) - (4xy + y^{2})\), then factor out the greatest common factor from each group to get \(3x(4x + y) - y(4x + y)\), and finally factor out the common binomial factor \((4x + y)\) to write the expression as \((3x - y)(4x + y)\).

Key Concepts

Factoring Trinomials

Factoring trinomials involves expressing a quadratic expression as a product of two binomials. For trinomials in the form ax^2 + bx + c, the goal is to find two binomials whose product equals the original trinomial. This process often requires identifying pairs of numbers that multiply to ac and add to b.

Handling Multiple Variables in Polynomials

When factoring polynomials with more than one variable, such as 12x^2 - xy - y^2, it is important to treat each variable carefully and consider their coefficients and powers. The variables can be factored similarly to single-variable polynomials, but attention must be paid to the terms involving different variables.

Using the AC Method (or Trial and Error) for Factoring

The AC method helps factor trinomials where the leading coefficient a is not 1. Multiply a and c, then find two numbers that multiply to ac and add to b. These numbers help split the middle term, allowing factoring by grouping. This method is useful for complex trinomials like 12x^2 - xy - y^2.