SOLUTION: Factor the following expression completely: 70w^3 – 125w^2 + 30w

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Question 550941: Factor the following expression completely:
70w^3 – 125w^2 + 30w
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

70w%5E3-125w%5E2%2B30w Start with the given expression.


5w%2814w%5E2-25w%2B6%29 Factor out the GCF 5w.


Now let's try to factor the inner expression 14w%5E2-25w%2B6


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Looking at the expression 14w%5E2-25w%2B6, we can see that the first coefficient is 14, the second coefficient is -25, and the last term is 6.


Now multiply the first coefficient 14 by the last term 6 to get %2814%29%286%29=84.


Now the question is: what two whole numbers multiply to 84 (the previous product) and add to the second coefficient -25?


To find these two numbers, we need to list all of the factors of 84 (the previous product).


Factors of 84:
1,2,3,4,6,7,12,14,21,28,42,84
-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84


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


These factors pair up and multiply to 84.
1*84 = 84
2*42 = 84
3*28 = 84
4*21 = 84
6*14 = 84
7*12 = 84
(-1)*(-84) = 84
(-2)*(-42) = 84
(-3)*(-28) = 84
(-4)*(-21) = 84
(-6)*(-14) = 84
(-7)*(-12) = 84

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -25:


First NumberSecond NumberSum
1841+84=85
2422+42=44
3283+28=31
4214+21=25
6146+14=20
7127+12=19
-1-84-1+(-84)=-85
-2-42-2+(-42)=-44
-3-28-3+(-28)=-31
-4-21-4+(-21)=-25
-6-14-6+(-14)=-20
-7-12-7+(-12)=-19



From the table, we can see that the two numbers -4 and -21 add to -25 (the middle coefficient).


So the two numbers -4 and -21 both multiply to 84 and add to -25


Now replace the middle term -25w with -4w-21w. Remember, -4 and -21 add to -25. So this shows us that -4w-21w=-25w.


14w%5E2%2Bhighlight%28-4w-21w%29%2B6 Replace the second term -25w with -4w-21w.


%2814w%5E2-4w%29%2B%28-21w%2B6%29 Group the terms into two pairs.


2w%287w-2%29%2B%28-21w%2B6%29 Factor out the GCF 2w from the first group.


2w%287w-2%29-3%287w-2%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.


%282w-3%29%287w-2%29 Combine like terms. Or factor out the common term 7w-2


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So 5w%2814w%5E2-25w%2B6%29 then factors further to 5w%282w-3%29%287w-2%29


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Answer:


So 70w%5E3-125w%5E2%2B30w completely factors to 5w%282w-3%29%287w-2%29.


In other words, 70w%5E3-125w%5E2%2B30w=5w%282w-3%29%287w-2%29.


Note: you can check the answer by expanding 5w%282w-3%29%287w-2%29 to get 70w%5E3-125w%5E2%2B30w or by graphing the original expression and the answer (the two graphs should be identical).