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Chapter 0: Problem 38

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

Short Answer

Expert verified
The factored form of the trinomial \( 6x^{2}-7xy-5y^{2} \) is \( (3x - 5y)(2x + y) \).

Step by step solution

01

Identify and Arrange Coefficients

Identify the coefficients a, b and c from the given trinomial: \( a = 6 \), \( b = -7 \), \( c = -5 \). The trinomial in this case is arranged in the form \( ax^{2}+ bx + cy^{2} \), which is \( 6x^{2} - 7xy - 5y^{2} \).
02

Find the pair of factors

The next step involves finding a pair of numbers such that their product equals the product of a and c and their sum equals the coefficient b. So, we need to find numbers that multiply to give \( ac = 6*-5 = -30 \) and add to give \( b = -7 \) . The numbers that fit this condition are -10 and 3 since \( -10 * 3 = -30 \) and \( -10 + 3 = -7 \).
03

Rewrite the equation by splitting the middle term

We can now rewrite the middle term of the trinomial using these factors: \( -7x = -10x + 3x \). This leads us to: \( 6x^{2} - 10xy + 3xy -5y^{2} \)
04

Apply grouping

Group the terms in pairs for factoring by group: \( (6x^{2}-10xy) + (3xy-5y^{2}) \).
05

Factor by grouping

Now, factor by grouping: \( 2x(3x-5y) + y(3x-5y) \).
06

Result

The final factored form of the given trinomial is obtained by taking out the common factor, which results in: \( (3x - 5y)(2x + y) \).

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