SOLUTION: u^2-2uv-63v^2 ----------- ---- x^2+2xy-24y^2 ----------- ---- u^2-2uv-8v^2 ----------- ---- u^2-2uv-8v^2 ----------- ---- u^2-3uv-40v^2 ----------- ---- x^2+50x+625 ----

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: u^2-2uv-63v^2 ----------- ---- x^2+2xy-24y^2 ----------- ---- u^2-2uv-8v^2 ----------- ---- u^2-2uv-8v^2 ----------- ---- u^2-3uv-40v^2 ----------- ---- x^2+50x+625 ----      Log On


   

Question 252211: u^2-2uv-63v^2
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x^2+2xy-24y^2
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u^2-2uv-8v^2
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u^2-2uv-8v^2
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u^2-3uv-40v^2
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x^2+50x+625
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x^2-8xy+16y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started.

# 1



Looking at u%5E2-2uv-63v%5E2 we can see that the first term is u%5E2 and the last term is -63v%5E2 where the coefficients are 1 and -63 respectively.

Now multiply the first coefficient 1 and the last coefficient -63 to get -63. Now what two numbers multiply to -63 and add to the middle coefficient -2? Let's list all of the factors of -63:



Factors of -63:
1,3,7,9,21,63

-1,-3,-7,-9,-21,-63 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -63
(1)*(-63)
(3)*(-21)
(7)*(-9)
(-1)*(63)
(-3)*(21)
(-7)*(9)

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


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First NumberSecond NumberSum
1-631+(-63)=-62
3-213+(-21)=-18
7-97+(-9)=-2
-163-1+63=62
-321-3+21=18
-79-7+9=2



From this list we can see that 7 and -9 add up to -2 and multiply to -63


Now looking at the expression u%5E2-2uv-63v%5E2, replace -2uv with 7uv-9uv (notice 7uv-9uv combines back to -2uv. So it is equivalent to -2uv)

u%5E2%2Bhighlight%287uv-9uv%29-63v%5E2


Now let's factor u%5E2%2B7uv-9uv-63v%5E2 by grouping:


%28u%5E2%2B7uv%29%2B%28-9uv-63v%5E2%29 Group like terms


u%28u%2B7v%29-9v%28u%2B7v%29 Factor out the GCF of u out of the first group. Factor out the GCF of -9v out of the second group


%28u-9v%29%28u%2B7v%29 Since we have a common term of u%2B7v, we can combine like terms

So u%5E2%2B7uv-9uv-63v%5E2 factors to %28u-9v%29%28u%2B7v%29


So this also means that u%5E2-2uv-63v%5E2 factors to %28u-9v%29%28u%2B7v%29 (since u%5E2-2uv-63v%5E2 is equivalent to u%5E2%2B7uv-9uv-63v%5E2)



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Answer:
So u%5E2-2uv-63v%5E2 factors to %28u-9v%29%28u%2B7v%29


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# 2



Looking at x%5E2%2B2xy-24y%5E2 we can see that the first term is x%5E2 and the last term is -24y%5E2 where the coefficients are 1 and -24 respectively.

Now multiply the first coefficient 1 and the last coefficient -24 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient 2? Let's list all of the factors of -24:



Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

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


Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From this list we can see that -4 and 6 add up to 2 and multiply to -24


Now looking at the expression x%5E2%2B2xy-24y%5E2, replace 2xy with -4xy%2B6xy (notice -4xy%2B6xy combines back to 2xy. So it is equivalent to 2xy)

x%5E2%2Bhighlight%28-4xy%2B6xy%29-24y%5E2


Now let's factor x%5E2-4xy%2B6xy-24y%5E2 by grouping:


%28x%5E2-4xy%29%2B%286xy-24y%5E2%29 Group like terms


x%28x-4y%29%2B6y%28x-4y%29 Factor out the GCF of x out of the first group. Factor out the GCF of 6y out of the second group


%28x%2B6y%29%28x-4y%29 Since we have a common term of x-4y, we can combine like terms

So x%5E2-4xy%2B6xy-24y%5E2 factors to %28x%2B6y%29%28x-4y%29


So this also means that x%5E2%2B2xy-24y%5E2 factors to %28x%2B6y%29%28x-4y%29 (since x%5E2%2B2xy-24y%5E2 is equivalent to x%5E2-4xy%2B6xy-24y%5E2)



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Answer:
So x%5E2%2B2xy-24y%5E2 factors to %28x%2B6y%29%28x-4y%29


The rest of the problems are all similar to these two. So I'll let you finish up.