Problem 38 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-7 x y-5 y^{2}$$

Step-by-Step Solution

Answer

The factors of the given trinomial \(6 x^{2}-7 x y-5 y^{2}\) are \(-2y + 3x\) and \(-3y + 2x\).

Step 1 - Identify the Form of the Given Trinomial

The trinomial given, \(6 x^{2}-7 x y-5 y^{2}\), has the standard form \(ax^2 + bxy + cy^2\). Here, \(a = 6\), \(b = -7\), and \(c = -5\).

Step 2 - Guess and Check

The next step is to make informed guesses about what the factors of the trinomial using the relations \(a = r*s\), \(ac = r*t*s*u\) and \(b = r*u + t*s\). Factor the coefficients a and c and try the subtraction or addition combinations to get b. After some trial and error, one might note that \(r = -2y\), \(s = -3y\), \(t = 3x\), and \(u = 2x\) are the factors satisfying both the conditions.

Step 3 - Write down the Factors

After finding the correct coefficients for \(r, s, t\) and \(u\), the required factors of the given expression are \(-2y + 3x\) and \(-3y + 2x\).