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Chapter 0: Problem 36

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

Short Answer

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

Step by step solution

01

Find two numbers to split the middle term

Find two numbers that multiply to \(-30\) (i.e., from \(6 x -5\)) and add up to \(-7\) (i.e., from \(-7 x y\)). The numbers are \(10\) and \(-3\).
02

Split the middle term

Replace \(-7 x y\) in the given trinomial with \(10 x y - 3 x y\), to obtain \(6 x^2 + 10 x y - 3 x y - 5 y^2\).
03

Group and factor

Rearrange the trinomial into two pairs \((6 x^2 + 10 x y)\) and \((-3 x y - 5 y^2)\) and factor the greatest common factor from each pair, to get \(2 x (3 x + 5 y) - y (3 x + 5 y)\).
04

Achievement of factorization

Both resulting terms share a common binomial factor \((3 x + 5 y)\), which leads to the final factorized form by grouping \((3 x + 5 y)\) together: \((2 x - y)(3 x + 5 y)\).

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