SOLUTION: How do you factor r^3+3r^2-54r

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Question 367308: How do you factor r^3+3r^2-54r
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

r%5E3%2B3r%5E2-54r Start with the given expression.


r%28r%5E2%2B3r-54%29 Factor out the GCF r.


Now let's try to factor the inner expression r%5E2%2B3r-54


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Looking at the expression r%5E2%2B3r-54, we can see that the first coefficient is 1, the second coefficient is 3, and the last term is -54.


Now multiply the first coefficient 1 by the last term -54 to get %281%29%28-54%29=-54.


Now the question is: what two whole numbers multiply to -54 (the previous product) and add to the second coefficient 3?


To find these two numbers, we need to list all of the factors of -54 (the previous product).


Factors of -54:
1,2,3,6,9,18,27,54
-1,-2,-3,-6,-9,-18,-27,-54


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


These factors pair up and multiply to -54.
1*(-54) = -54
2*(-27) = -54
3*(-18) = -54
6*(-9) = -54
(-1)*(54) = -54
(-2)*(27) = -54
(-3)*(18) = -54
(-6)*(9) = -54

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:


First NumberSecond NumberSum
1-541+(-54)=-53
2-272+(-27)=-25
3-183+(-18)=-15
6-96+(-9)=-3
-154-1+54=53
-227-2+27=25
-318-3+18=15
-69-6+9=3



From the table, we can see that the two numbers -6 and 9 add to 3 (the middle coefficient).


So the two numbers -6 and 9 both multiply to -54 and add to 3


Now replace the middle term 3r with -6r%2B9r. Remember, -6 and 9 add to 3. So this shows us that -6r%2B9r=3r.


r%5E2%2Bhighlight%28-6r%2B9r%29-54 Replace the second term 3r with -6r%2B9r.


%28r%5E2-6r%29%2B%289r-54%29 Group the terms into two pairs.


r%28r-6%29%2B%289r-54%29 Factor out the GCF r from the first group.


r%28r-6%29%2B9%28r-6%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%2B9%29%28r-6%29 Combine like terms. Or factor out the common term r-6


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So r%28r%5E2%2B3r-54%29 then factors further to r%28r%2B9%29%28r-6%29


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Answer:


So r%5E3%2B3r%5E2-54r completely factors to r%28r%2B9%29%28r-6%29.


In other words, r%5E3%2B3r%5E2-54r=r%28r%2B9%29%28r-6%29.


Note: you can check the answer by expanding r%28r%2B9%29%28r-6%29 to get r%5E3%2B3r%5E2-54r or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim