SOLUTION: Factor the expressions: -r^2+11r-18

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Question 908042: Factor the expressions:
-r^2+11r-18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

-r%5E2%2B11r-18 Start with the given expression.


-%28r%5E2-11r%2B18%29 Factor out the GCF -1.


Now let's try to factor the inner expression r%5E2-11r%2B18


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Looking at the expression r%5E2-11r%2B18, we can see that the first coefficient is 1, the second coefficient is -11, and the last term is 18.


Now multiply the first coefficient 1 by the last term 18 to get %281%29%2818%29=18.


Now the question is: what two whole numbers multiply to 18 (the previous product) and add to the second coefficient -11?


To find these two numbers, we need to list all of the factors of 18 (the previous product).


Factors of 18:
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18


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


These factors pair up and multiply to 18.
1*18 = 18
2*9 = 18
3*6 = 18
(-1)*(-18) = 18
(-2)*(-9) = 18
(-3)*(-6) = 18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -11:


First NumberSecond NumberSum
1181+18=19
292+9=11
363+6=9
-1-18-1+(-18)=-19
-2-9-2+(-9)=-11
-3-6-3+(-6)=-9



From the table, we can see that the two numbers -2 and -9 add to -11 (the middle coefficient).


So the two numbers -2 and -9 both multiply to 18 and add to -11


Now replace the middle term -11r with -2r-9r. Remember, -2 and -9 add to -11. So this shows us that -2r-9r=-11r.


r%5E2%2Bhighlight%28-2r-9r%29%2B18 Replace the second term -11r with -2r-9r.


%28r%5E2-2r%29%2B%28-9r%2B18%29 Group the terms into two pairs.


r%28r-2%29%2B%28-9r%2B18%29 Factor out the GCF r from the first group.


r%28r-2%29-9%28r-2%29 Factor out 9 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-9%29%28r-2%29 Combine like terms. Or factor out the common term r-2


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So -1%28r%5E2-11r%2B18%29 then factors further to -%28r-9%29%28r-2%29


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Answer:


So -r%5E2%2B11r-18 completely factors to -%28r-9%29%28r-2%29.


In other words, -r%5E2%2B11r-18=-%28r-9%29%28r-2%29.


Note: you can check the answer by expanding -%28r-9%29%28r-2%29 to get -r%5E2%2B11r-18 or by graphing the original expression and the answer (the two graphs should be identical).