SOLUTION: Factor and check: x^2 - 3xy - 4y^2

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Question 146173This question is from textbook
: Factor and check: x^2 - 3xy - 4y^2 This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at x%5E2-3xy-4y%5E2 we can see that the first term is x%5E2 and the last term is -4y%5E2 where the coefficients are 1 and -4 respectively.

Now multiply the first coefficient 1 and the last coefficient -4 to get -4. Now what two numbers multiply to -4 and add to the middle coefficient -3? Let's list all of the factors of -4:



Factors of -4:
1,2

-1,-2 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -4
(1)*(-4)
(-1)*(4)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -3

First NumberSecond NumberSum
1-41+(-4)=-3
-14-1+4=3



From this list we can see that 1 and -4 add up to -3 and multiply to -4


Now looking at the expression x%5E2-3xy-4y%5E2, replace -3xy with xy%2B-4xy (notice xy%2B-4xy adds up to -3xy. So it is equivalent to -3xy)

x%5E2%2Bhighlight%28xy%2B-4xy%29%2B-4y%5E2


Now let's factor x%5E2%2Bxy-4xy-4y%5E2 by grouping:


%28x%5E2%2Bxy%29%2B%28-4xy-4y%5E2%29 Group like terms


x%28x%2By%29-4y%28x%2By%29 Factor out the GCF of x out of the first group. Factor out the GCF of -4y out of the second group


%28x-4y%29%28x%2By%29 Since we have a common term of x%2By, we can combine like terms

So x%5E2%2Bxy-4xy-4y%5E2 factors to %28x-4y%29%28x%2By%29


So this also means that x%5E2-3xy-4y%5E2 factors to %28x-4y%29%28x%2By%29 (since x%5E2-3xy-4y%5E2 is equivalent to x%5E2%2Bxy-4xy-4y%5E2)



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Answer:
So x%5E2-3xy-4y%5E2 factors to %28x-4y%29%28x%2By%29



Check:


In order to check our answer, we need to FOIL %28x-4y%29%28x%2By%29


Remember, when you FOIL an expression, you follow this procedure:


%28highlight%28x%29-4y%29%28highlight%28x%29%2By%29 Multiply the First terms:%28x%29%2A%28x%29=x%5E2.


%28highlight%28x%29-4y%29%28x%2Bhighlight%28y%29%29 Multiply the Outer terms:%28x%29%2A%28y%29=x%2Ay.


%28x%2Bhighlight%28-4y%29%29%28highlight%28x%29%2By%29 Multiply the Inner terms:%28-4%2Ay%29%2A%28x%29=-4%2Ax%2Ay.


%28x%2Bhighlight%28-4y%29%29%28x%2Bhighlight%28y%29%29 Multiply the Last terms:%28-4%2Ay%29%2A%28y%29=-4%2Ay%5E2.


x%5E2%2Bx%2Ay-4%2Ax%2Ay-4%2Ay%5E2 Now collect every term to make a single expression.


x%5E2-3%2Ax%2Ay-4%2Ay%5E2 Now combine like terms.


So %28x-4y%29%28x%2By%29 FOILS to x%5E2-3%2Ax%2Ay-4%2Ay%5E2.


In other words, %28x-4y%29%28x%2By%29=x%5E2-3%2Ax%2Ay-4%2Ay%5E2.


So this verifies our answer.