SOLUTION: How do you factor 25r^2+10r+1

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Question 473308: How do you factor 25r^2+10r+1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 25r%5E2%2B10r%2B1, we can see that the first coefficient is 25, the second coefficient is 10, and the last term is 1.


Now multiply the first coefficient 25 by the last term 1 to get %2825%29%281%29=25.


Now the question is: what two whole numbers multiply to 25 (the previous product) and add to the second coefficient 10?


To find these two numbers, we need to list all of the factors of 25 (the previous product).


Factors of 25:
1,5,25
-1,-5,-25


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


These factors pair up and multiply to 25.
1*25 = 25
5*5 = 25
(-1)*(-25) = 25
(-5)*(-5) = 25

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:


First NumberSecond NumberSum
1251+25=26
555+5=10
-1-25-1+(-25)=-26
-5-5-5+(-5)=-10



From the table, we can see that the two numbers 5 and 5 add to 10 (the middle coefficient).


So the two numbers 5 and 5 both multiply to 25 and add to 10


Now replace the middle term 10r with 5r%2B5r. Remember, 5 and 5 add to 10. So this shows us that 5r%2B5r=10r.


25r%5E2%2Bhighlight%285r%2B5r%29%2B1 Replace the second term 10r with 5r%2B5r.


%2825r%5E2%2B5r%29%2B%285r%2B1%29 Group the terms into two pairs.


5r%285r%2B1%29%2B%285r%2B1%29 Factor out the GCF 5r from the first group.


5r%285r%2B1%29%2B1%285r%2B1%29 Factor out 1 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.


%285r%2B1%29%285r%2B1%29 Combine like terms. Or factor out the common term 5r%2B1


%285r%2B1%29%5E2 Condense the terms.


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Answer:


So 25r%5E2%2B10r%2B1 factors to %285r%2B1%29%5E2.


In other words, 25r%5E2%2B10r%2B1=%285r%2B1%29%5E2.


Note: you can check the answer by expanding %285r%2B1%29%5E2 to get 25r%5E2%2B10r%2B1 or by graphing the original expression and the answer (the two graphs should be identical).