SOLUTION: Make sure to superscript all exponents. If you believe that a polynomial is not factorable, label it as prime, Show all stages of factoring separately(i.e., show the result of fact

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Question 176141: Make sure to superscript all exponents. If you believe that a polynomial is not factorable, label it as prime, Show all stages of factoring separately(i.e., show the result of factoring out a GCF before taking the resulting polynomial and factoring it again).
2x^3-26x^2+80x

Thanks

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




------------------------------------------------------------



Looking at we can see that the first term is and the last term is where the coefficients are 1 and 40 respectively.

Now multiply the first coefficient 1 and the last coefficient 40 to get 40. Now what two numbers multiply to 40 and add to the middle coefficient -13? Let's list all of the factors of 40:



Factors of 40:
1,2,4,5,8,10,20,40

-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 40
1*40
2*20
4*10
5*8
(-1)*(-40)
(-2)*(-20)
(-4)*(-10)
(-5)*(-8)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13

First NumberSecond NumberSum
1401+40=41
2202+20=22
4104+10=14
585+8=13
-1-40-1+(-40)=-41
-2-20-2+(-20)=-22
-4-10-4+(-10)=-14
-5-8-5+(-8)=-13



From this list we can see that -5 and -8 add up to -13 and multiply to 40


Now looking at the expression , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )



------------------------------------------------------------




So our expression goes from and factors further to


------------------
Answer:

So completely factors to

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