SOLUTION: Factor 81x^2-126xy+49y^2
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Question 610312: Factor 81x^2-126xy+49y^2
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last coefficient is .
Now multiply the first coefficient by the last coefficient to get .
Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,3,7,9,21,27,49,63,81,147,189,441,567,1323,3969
-1,-3,-7,-9,-21,-27,-49,-63,-81,-147,-189,-441,-567,-1323,-3969
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*3969 = 3969
3*1323 = 3969
7*567 = 3969
9*441 = 3969
21*189 = 3969
27*147 = 3969
49*81 = 3969
63*63 = 3969
(-1)*(-3969) = 3969
(-3)*(-1323) = 3969
(-7)*(-567) = 3969
(-9)*(-441) = 3969
(-21)*(-189) = 3969
(-27)*(-147) = 3969
(-49)*(-81) = 3969
(-63)*(-63) = 3969
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 3969 | 1+3969=3970 |
| 3 | 1323 | 3+1323=1326 |
| 7 | 567 | 7+567=574 |
| 9 | 441 | 9+441=450 |
| 21 | 189 | 21+189=210 |
| 27 | 147 | 27+147=174 |
| 49 | 81 | 49+81=130 |
| 63 | 63 | 63+63=126 |
| -1 | -3969 | -1+(-3969)=-3970 |
| -3 | -1323 | -3+(-1323)=-1326 |
| -7 | -567 | -7+(-567)=-574 |
| -9 | -441 | -9+(-441)=-450 |
| -21 | -189 | -21+(-189)=-210 |
| -27 | -147 | -27+(-147)=-174 |
| -49 | -81 | -49+(-81)=-130 |
| -63 | -63 | -63+(-63)=-126 |
From the table, we can see that the two numbers and add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term with . Remember, and add to . So this shows us that .
Replace the second term with .
Group the terms into two pairs.
Factor out the GCF from the first group.
Factor out the from the second group.
Factor out the from the entire expression.
===============================================================
Answer:
So completely factors to
In other words, for all values of x and y.
Note: you can check the answer by expanding to get back again or by graphing the original expression and the answer (the two graphs should be identical).
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