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Chapter 0: Problem 37

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

Short Answer

Expert verified
The factorization of the trinomial \(6x^2 - 5xy - 6y^2\) is \((3x-2y)(2x-3y)\)

Step by step solution

01

Identify values of 'a', 'b' and 'c'

From the given equation \(6x^2 - 5xy - 6y^2\), we identify \(a = 6\), \(b = -5\) and \(c = -6\) where 'a' is the coefficient of \(x^2\), 'b' is the coefficient of 'x' and 'c' is the constant term. Here 'b' also includes the term 'y'.
02

Find product and sum

Here, the product 'ac' would be \(6*(-6) = -36\) and the sum would be the coefficient of 'x' which is '-5'.
03

Factorization

Now, we need to find two numbers whose product is -36 (obtained from the 'ac' above) and whose sum is -5. From the numbers that multiply to -36, namely (-1, 36), (1, -36), (-2, 18), (2, -18), (-3, 12), (3, -12), (-4, 9), (4, -9) and (6, -6), we can see that (-9, 4) sums up to give -5.
04

Rewrite middle term

We write the equation obtained from the above steps by replacing the middle term with the number we found in step 3: \(6x^2 - 9xy + 4xy - 6y^2\).
05

Group and Factorize

Now, group the equation as follow: \(6x^2 - 9xy\) and \(4xy - 6y^2\), so we can begin factoring by common factor. For the first group, factor out '3x' and for the second group, factor out '2y': \(3x(2x-3y) - 2y(2x-3y)\)
06

Final factorization

Now, factor out the common binomial \((2x-3y)\)to get the final answer: \((3x-2y)(2x-3y)\)

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Most popular questions from this chapter

When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Exercises \(142-144\) will help you prepare for the material covered in the next section. $$\text { Multiply: }\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
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