SOLUTION: Factor completely: 40x^3 + 34x^2 - 20x

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Question 610032: Factor completely:
40x^3 + 34x^2 - 20x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

40x%5E3%2B34x%5E2-20x Start with the given expression.


2x%2820x%5E2%2B17x-10%29 Factor out the GCF 2x.


Now let's try to factor the inner expression 20x%5E2%2B17x-10


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Looking at the expression 20x%5E2%2B17x-10, we can see that the first coefficient is 20, the second coefficient is 17, and the last term is -10.


Now multiply the first coefficient 20 by the last term -10 to get %2820%29%28-10%29=-200.


Now the question is: what two whole numbers multiply to -200 (the previous product) and add to the second coefficient 17?


To find these two numbers, we need to list all of the factors of -200 (the previous product).


Factors of -200:
1,2,4,5,8,10,20,25,40,50,100,200
-1,-2,-4,-5,-8,-10,-20,-25,-40,-50,-100,-200


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


These factors pair up and multiply to -200.
1*(-200) = -200
2*(-100) = -200
4*(-50) = -200
5*(-40) = -200
8*(-25) = -200
10*(-20) = -200
(-1)*(200) = -200
(-2)*(100) = -200
(-4)*(50) = -200
(-5)*(40) = -200
(-8)*(25) = -200
(-10)*(20) = -200

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 17:


First NumberSecond NumberSum
1-2001+(-200)=-199
2-1002+(-100)=-98
4-504+(-50)=-46
5-405+(-40)=-35
8-258+(-25)=-17
10-2010+(-20)=-10
-1200-1+200=199
-2100-2+100=98
-450-4+50=46
-540-5+40=35
-825-8+25=17
-1020-10+20=10



From the table, we can see that the two numbers -8 and 25 add to 17 (the middle coefficient).


So the two numbers -8 and 25 both multiply to -200 and add to 17


Now replace the middle term 17x with -8x%2B25x. Remember, -8 and 25 add to 17. So this shows us that -8x%2B25x=17x.


20x%5E2%2Bhighlight%28-8x%2B25x%29-10 Replace the second term 17x with -8x%2B25x.


%2820x%5E2-8x%29%2B%2825x-10%29 Group the terms into two pairs.


4x%285x-2%29%2B%2825x-10%29 Factor out the GCF 4x from the first group.


4x%285x-2%29%2B5%285x-2%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.


%284x%2B5%29%285x-2%29 Combine like terms. Or factor out the common term 5x-2


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So 2x%2820x%5E2%2B17x-10%29 then factors further to 2x%284x%2B5%29%285x-2%29


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Answer:


So 40x%5E3%2B34x%5E2-20x completely factors to 2x%284x%2B5%29%285x-2%29.


In other words, 40x%5E3%2B34x%5E2-20x=2x%284x%2B5%29%285x-2%29.


Note: you can check the answer by expanding 2x%284x%2B5%29%285x-2%29 to get 40x%5E3%2B34x%5E2-20x or by graphing the original expression and the answer (the two graphs should be identical).

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