SOLUTION: factor 24w^4-20w^3-16w^2

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Question 704991: factor 24w^4-20w^3-16w^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
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24w%5E4-20w%5E3-16w%5E2 Start with the given expression.


4w%5E2%286w%5E2-5w-4%29 Factor out the GCF 4w%5E2.


Now let's try to factor the inner expression 6w%5E2-5w-4


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Looking at the expression 6w%5E2-5w-4, we can see that the first coefficient is 6, the second coefficient is -5, and the last term is -4.


Now multiply the first coefficient 6 by the last term -4 to get %286%29%28-4%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


Factors of -24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


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


These factors pair up and multiply to -24.
1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers 3 and -8 add to -5 (the middle coefficient).


So the two numbers 3 and -8 both multiply to -24 and add to -5


Now replace the middle term -5w with 3w-8w. Remember, 3 and -8 add to -5. So this shows us that 3w-8w=-5w.


6w%5E2%2Bhighlight%283w-8w%29-4 Replace the second term -5w with 3w-8w.


%286w%5E2%2B3w%29%2B%28-8w-4%29 Group the terms into two pairs.


3w%282w%2B1%29%2B%28-8w-4%29 Factor out the GCF 3w from the first group.


3w%282w%2B1%29-4%282w%2B1%29 Factor out 4 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.


%283w-4%29%282w%2B1%29 Combine like terms. Or factor out the common term 2w%2B1


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So 4w%5E2%286w%5E2-5w-4%29 then factors further to 4w%5E2%283w-4%29%282w%2B1%29


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Answer:


So 24w%5E4-20w%5E3-16w%5E2 completely factors to 4w%5E2%283w-4%29%282w%2B1%29.


In other words, 24w%5E4-20w%5E3-16w%5E2=4w%5E2%283w-4%29%282w%2B1%29.


Note: you can check the answer by expanding 4w%5E2%283w-4%29%282w%2B1%29 to get 24w%5E4-20w%5E3-16w%5E2 or by graphing the original expression and the answer (the two graphs should be identical).

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Jim

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