Universitario

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 factorsof  (the previous product). 

Factors of : 
1,2,3,5,6,10,15,30 
-1,-2,-3,-5,-6,-10,-15,-30 

Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 
1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30 

Now let's add up eachpair of factors to see if one pair adds to the middle coefficient : 

First Number | Second Number | Sum |
1 | -30 | 1+(-30)=-29 |
2 | -15 | 2+(-15)=-13 |
3 | -10 | 3+(-10)=-7 |
5| -6 | 5+(-6)=-1 |
-1 | 30 | -1+30=29 |
-2 | 15 | -2+15=13 |
-3 | 10 | -3+10=7 |
-5 | 6 | -5+6=1 |

From the table, we can see that the two numbers  and  add to  (the middlecoefficient). 

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 secondterm  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 parenthesisequal to the terms in the first parenthesis. 

 Combine like terms. Or factor out the common term  

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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).