Question 960518: How do you factor this special case polynomial?
4d squared + 36d + 81
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 term is  .
Now multiply the first coefficient by the last term 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 :
1,2,3,4,6,9,12,18,27,36,54,81,108,162,324
-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162,-324
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*324 = 324
2*162 = 324
3*108 = 324
4*81 = 324
6*54 = 324
9*36 = 324
12*27 = 324
18*18 = 324
(-1)*(-324) = 324
(-2)*(-162) = 324
(-3)*(-108) = 324
(-4)*(-81) = 324
(-6)*(-54) = 324
(-9)*(-36) = 324
(-12)*(-27) = 324
(-18)*(-18) = 324
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 | 324 | 1+324=325 | | 2 | 162 | 2+162=164 | | 3 | 108 | 3+108=111 | | 4 | 81 | 4+81=85 | | 6 | 54 | 6+54=60 | | 9 | 36 | 9+36=45 | | 12 | 27 | 12+27=39 | | 18 | 18 | 18+18=36 | | -1 | -324 | -1+(-324)=-325 | | -2 | -162 | -2+(-162)=-164 | | -3 | -108 | -3+(-108)=-111 | | -4 | -81 | -4+(-81)=-85 | | -6 | -54 | -6+(-54)=-60 | | -9 | -36 | -9+(-36)=-45 | | -12 | -27 | -12+(-27)=-39 | | -18 | -18 | -18+(-18)=-36 |
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  from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.
 Combine like terms. Or factor out the common term 
 Condense the terms.
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Answer:
So factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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