Problem 37 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-5 x y-6 y^{2}$$

Step-by-Step Solution

Answer

The factored form of \(6x^2 - 5xy - 6y^2\) is \((x - y)(6x + y)\)

Step 1: Checking If the Trinomial Can Be Factored

Find two numbers that multiply to -36 and add to -5. The numbers -6 and 6 satisfy these conditions. Thus, the trinomial can be factored.

Step 2: Apply Grouping

Rewrite the middle term of the trinomial as \(-6xy + xy\). Now, the expression becomes \(6x^2 - 6xy + xy - 6y^2\). Group the terms two by two, giving two groups \(6x^2 - 6xy\) and \(+ xy - 6y^2\).

Step 3: Factor Common Factors from Each Group

From the first group: \(6x^2 - 6xy = 6x(x - y)\). From the second group: \(xy - 6y^2 = y(x - y)\). So the original trinomial becomes \(6x(x - y) + y(x - y)\).

Step 4: Factor Common Factors Again

In the expression \(6x(x - y) + y(x - y)\), notice that we have a common factor of \(x - y\). Factor this out, giving \((x - y)(6x + y)\), which are the two binomials.