SOLUTION: Factor the trinomial 6r^2 +29r -42

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Question 473223: Factor the trinomial


6r^2 +29r -42
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression 6r%5E2%2B29r-42, we can see that the first coefficient is 6, the second coefficient is 29, and the last term is -42.


Now multiply the first coefficient 6 by the last term -42 to get %286%29%28-42%29=-252.


Now the question is: what two whole numbers multiply to -252 (the previous product) and add to the second coefficient 29?


To find these two numbers, we need to list all of the factors of -252 (the previous product).


Factors of -252:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -252.
1*(-252) = -252
2*(-126) = -252
3*(-84) = -252
4*(-63) = -252
6*(-42) = -252
7*(-36) = -252
9*(-28) = -252
12*(-21) = -252
14*(-18) = -252
(-1)*(252) = -252
(-2)*(126) = -252
(-3)*(84) = -252
(-4)*(63) = -252
(-6)*(42) = -252
(-7)*(36) = -252
(-9)*(28) = -252
(-12)*(21) = -252
(-14)*(18) = -252

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 29:


First NumberSecond NumberSum
1-2521+(-252)=-251
2-1262+(-126)=-124
3-843+(-84)=-81
4-634+(-63)=-59
6-426+(-42)=-36
7-367+(-36)=-29
9-289+(-28)=-19
12-2112+(-21)=-9
14-1814+(-18)=-4
-1252-1+252=251
-2126-2+126=124
-384-3+84=81
-463-4+63=59
-642-6+42=36
-736-7+36=29
-928-9+28=19
-1221-12+21=9
-1418-14+18=4



From the table, we can see that the two numbers -7 and 36 add to 29 (the middle coefficient).


So the two numbers -7 and 36 both multiply to -252 and add to 29


Now replace the middle term 29r with -7r%2B36r. Remember, -7 and 36 add to 29. So this shows us that -7r%2B36r=29r.


6r%5E2%2Bhighlight%28-7r%2B36r%29-42 Replace the second term 29r with -7r%2B36r.


%286r%5E2-7r%29%2B%2836r-42%29 Group the terms into two pairs.


r%286r-7%29%2B%2836r-42%29 Factor out the GCF r from the first group.


r%286r-7%29%2B6%286r-7%29 Factor out 6 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.


%28r%2B6%29%286r-7%29 Combine like terms. Or factor out the common term 6r-7


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Answer:


So 6r%5E2%2B29r-42 factors to %28r%2B6%29%286r-7%29.


In other words, 6r%5E2%2B29r-42=%28r%2B6%29%286r-7%29.


Note: you can check the answer by expanding %28r%2B6%29%286r-7%29 to get 6r%5E2%2B29r-42 or by graphing the original expression and the answer (the two graphs should be identical).