SOLUTION: factor the trinomial 24r^2-6r-45

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Question 615983: factor the trinomial 24r^2-6r-45
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

24r%5E2-6r-45 Start with the given expression.


3%288r%5E2-2r-15%29 Factor out the GCF 3.


Now let's try to factor the inner expression 8r%5E2-2r-15


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Looking at the expression 8r%5E2-2r-15, we can see that the first coefficient is 8, the second coefficient is -2, and the last term is -15.


Now multiply the first coefficient 8 by the last term -15 to get %288%29%28-15%29=-120.


Now the question is: what two whole numbers multiply to -120 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of -120 (the previous product).


Factors of -120:
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120


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


These factors pair up and multiply to -120.
1*(-120) = -120
2*(-60) = -120
3*(-40) = -120
4*(-30) = -120
5*(-24) = -120
6*(-20) = -120
8*(-15) = -120
10*(-12) = -120
(-1)*(120) = -120
(-2)*(60) = -120
(-3)*(40) = -120
(-4)*(30) = -120
(-5)*(24) = -120
(-6)*(20) = -120
(-8)*(15) = -120
(-10)*(12) = -120

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:


First NumberSecond NumberSum
1-1201+(-120)=-119
2-602+(-60)=-58
3-403+(-40)=-37
4-304+(-30)=-26
5-245+(-24)=-19
6-206+(-20)=-14
8-158+(-15)=-7
10-1210+(-12)=-2
-1120-1+120=119
-260-2+60=58
-340-3+40=37
-430-4+30=26
-524-5+24=19
-620-6+20=14
-815-8+15=7
-1012-10+12=2



From the table, we can see that the two numbers 10 and -12 add to -2 (the middle coefficient).


So the two numbers 10 and -12 both multiply to -120 and add to -2


Now replace the middle term -2r with 10r-12r. Remember, 10 and -12 add to -2. So this shows us that 10r-12r=-2r.


8r%5E2%2Bhighlight%2810r-12r%29-15 Replace the second term -2r with 10r-12r.


%288r%5E2%2B10r%29%2B%28-12r-15%29 Group the terms into two pairs.


2r%284r%2B5%29%2B%28-12r-15%29 Factor out the GCF 2r from the first group.


2r%284r%2B5%29-3%284r%2B5%29 Factor out 3 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.


%282r-3%29%284r%2B5%29 Combine like terms. Or factor out the common term 4r%2B5


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So 3%288r%5E2-2r-15%29 then factors further to 3%282r-3%29%284r%2B5%29


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Answer:


So 24r%5E2-6r-45 completely factors to 3%282r-3%29%284r%2B5%29.


In other words, 24r%5E2-6r-45=3%282r-3%29%284r%2B5%29.


Note: you can check the answer by expanding 3%282r-3%29%284r%2B5%29 to get 24r%5E2-6r-45 or by graphing the original expression and the answer (the two graphs should be identical).