SOLUTION: PLEASE SOMEONE COME TO MY RESCUE! How do I solve these kinds of problems:Factoring I do not understand how to figure out which number is the correct one. Does anyone know how to

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: PLEASE SOMEONE COME TO MY RESCUE! How do I solve these kinds of problems:Factoring I do not understand how to figure out which number is the correct one. Does anyone know how to      Log On


   

Question 590784: PLEASE SOMEONE COME TO MY RESCUE!
How do I solve these kinds of problems:Factoring
I do not understand how to figure out which number is the correct one.
Does anyone know how to work these and explain the process to me in a simple way that my brain can understand?
I cannot seem to grasp the concept.
Factor:
20+ 22r - 12r^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
20%2B+22r+-+12r%5E2 Start with the given expression.


-12r%5E2%2B22r%2B20 Rearrange the terms in descending exponent order.


-2%286r%5E2-11r-10%29 Factor out the GCF -2.


Now let's try to factor the inner expression 6r%5E2-11r-10


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Looking at the expression 6r%5E2-11r-10, we can see that the first coefficient is 6, the second coefficient is -11, and the last term is -10.


Now multiply the first coefficient 6 by the last term -10 to get %286%29%28-10%29=-60.


Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient -11?


To find these two numbers, we need to list all of the factors of -60 (the previous product).


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


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


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

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


First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4



From the table, we can see that the two numbers 4 and -15 add to -11 (the middle coefficient).


So the two numbers 4 and -15 both multiply to -60 and add to -11


Now replace the middle term -11r with 4r-15r. Remember, 4 and -15 add to -11. So this shows us that 4r-15r=-11r.


6r%5E2%2Bhighlight%284r-15r%29-10 Replace the second term -11r with 4r-15r.


%286r%5E2%2B4r%29%2B%28-15r-10%29 Group the terms into two pairs.


2r%283r%2B2%29%2B%28-15r-10%29 Factor out the GCF 2r from the first group.


2r%283r%2B2%29-5%283r%2B2%29 Factor out 5 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-5%29%283r%2B2%29 Combine like terms. Or factor out the common term 3r%2B2


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So -2%286r%5E2-11r-10%29 then factors further to -2%282r-5%29%283r%2B2%29


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Answer:


So 20%2B+22r+-+12r%5E2 completely factors to -2%282r-5%29%283r%2B2%29.


In other words, 20%2B+22r+-+12r%5E2=-2%282r-5%29%283r%2B2%29.


Note: you can check the answer by expanding -2%282r-5%29%283r%2B2%29 to get 20%2B+22r+-+12r%5E2 or by graphing the original expression and the answer (the two graphs should be identical).
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