SOLUTION: Completly factor the polynominal h^4+2h^3-8h^2 I dont even know where to start on this one.. so I dont have any work to show :(

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Question 215064: Completly factor the polynominal h^4+2h^3-8h^2
I dont even know where to start on this one.. so I dont have any work to show :(
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


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h%5E4%2B2h%5E3-8h%5E2 Start with the given expression.


h%5E2%28h%5E2%2B2h-8%29 Factor out the GCF h%5E2.


Now let's try to factor the inner expression h%5E2%2B2h-8


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Looking at the expression h%5E2%2B2h-8, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -8.


Now multiply the first coefficient 1 by the last term -8 to get %281%29%28-8%29=-8.


Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -8 (the previous product).


Factors of -8:
1,2,4,8
-1,-2,-4,-8


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2



From the table, we can see that the two numbers -2 and 4 add to 2 (the middle coefficient).


So the two numbers -2 and 4 both multiply to -8 and add to 2


Now replace the middle term 2h with -2h%2B4h. Remember, -2 and 4 add to 2. So this shows us that -2h%2B4h=2h.


h%5E2%2Bhighlight%28-2h%2B4h%29-8 Replace the second term 2h with -2h%2B4h.


%28h%5E2-2h%29%2B%284h-8%29 Group the terms into two pairs.


h%28h-2%29%2B%284h-8%29 Factor out the GCF h from the first group.


h%28h-2%29%2B4%28h-2%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.


%28h%2B4%29%28h-2%29 Combine like terms. Or factor out the common term h-2


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So h%5E2%28h%5E2%2B2h-8%29 then factors further to h%5E2%28h%2B4%29%28h-2%29


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Answer:


So h%5E4%2B2h%5E3-8h%5E2 completely factors to h%5E2%28h%2B4%29%28h-2%29.


In other words, h%5E4%2B2h%5E3-8h%5E2=h%5E2%28h%2B4%29%28h-2%29.


Note: you can check the answer by expanding h%5E2%28h%2B4%29%28h-2%29 to get h%5E4%2B2h%5E3-8h%5E2 or by graphing the original expression and the answer (the two graphs should be identical).

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