SOLUTION: HOW DO I FACTOR PROBLEMS LIKE X^2-9XY+14Y^2 OR 6a^2-5ab-4b^2 please show the work for it thanks

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: HOW DO I FACTOR PROBLEMS LIKE X^2-9XY+14Y^2 OR 6a^2-5ab-4b^2 please show the work for it thanks      Log On


   

Question 195512: HOW DO I FACTOR PROBLEMS LIKE
X^2-9XY+14Y^2
OR
6a^2-5ab-4b^2
please show the work for it
thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



Looking at x%5E2-9xy%2B14y%5E2 we can see that the first term is x%5E2 and the last term is 14y%5E2 where the coefficients are 1 and 14 respectively.

Now multiply the first coefficient 1 and the last coefficient 14 to get 14. Now what two numbers multiply to 14 and add to the middle coefficient -9? Let's list all of the factors of 14:



Factors of 14:
1,2,7,14

-1,-2,-7,-14 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 14
1*14
2*7
(-1)*(-14)
(-2)*(-7)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9

First NumberSecond NumberSum
1141+14=15
272+7=9
-1-14-1+(-14)=-15
-2-7-2+(-7)=-9



From this list we can see that -2 and -7 add up to -9 and multiply to 14


Now looking at the expression x%5E2-9xy%2B14y%5E2, replace -9xy with -2xy-7xy (notice -2xy-7xy combines to -9xy. So it is equivalent to -9xy)

x%5E2%2Bhighlight%28-2xy-7xy%29%2B14y%5E2


Now let's factor x%5E2-2xy-7xy%2B14y%5E2 by grouping:


%28x%5E2-2xy%29%2B%28-7xy%2B14y%5E2%29 Group like terms


x%28x-2y%29-7y%28x-2y%29 Factor out the GCF of x out of the first group. Factor out the GCF of -7y out of the second group


%28x-7y%29%28x-2y%29 Since we have a common term of x-2y, we can combine like terms

So x%5E2-2xy-7xy%2B14y%5E2 factors to %28x-7y%29%28x-2y%29


So this also means that x%5E2-9xy%2B14y%5E2 factors to %28x-7y%29%28x-2y%29 (since x%5E2-9xy%2B14y%5E2 is equivalent to x%5E2-2xy-7xy%2B14y%5E2)



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Answer:
So x%5E2-9xy%2B14y%5E2 factors to %28x-7y%29%28x-2y%29




# 2



Looking at 6a%5E2-5ab-4b%5E2 we can see that the first term is 6a%5E2 and the last term is -4b%5E2 where the coefficients are 6 and -4 respectively.

Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? 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 -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

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 3 and -8 add up to -5 and multiply to -24


Now looking at the expression 6a%5E2-5ab-4b%5E2, replace -5ab with 3ab%2B-8ab (notice 3ab%2B-8ab adds up to -5ab. So it is equivalent to -5ab)

6a%5E2%2Bhighlight%283ab%2B-8ab%29%2B-4b%5E2


Now let's factor 6a%5E2%2B3ab-8ab-4b%5E2 by grouping:


%286a%5E2%2B3ab%29%2B%28-8ab-4b%5E2%29 Group like terms


3a%282a%2Bb%29-4b%282a%2Bb%29 Factor out the GCF of 3a out of the first group. Factor out the GCF of -4b out of the second group


%283a-4b%29%282a%2Bb%29 Since we have a common term of 2a%2Bb, we can combine like terms

So 6a%5E2%2B3ab-8ab-4b%5E2 factors to %283a-4b%29%282a%2Bb%29


So this also means that 6a%5E2-5ab-4b%5E2 factors to %283a-4b%29%282a%2Bb%29 (since 6a%5E2-5ab-4b%5E2 is equivalent to 6a%5E2%2B3ab-8ab-4b%5E2)



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Answer:
So 6a%5E2-5ab-4b%5E2 factors to %283a-4b%29%282a%2Bb%29