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

Algebra ->  Distributive-associative-commutative-properties -> 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      Log On


   

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

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

2x%5E3-26x%5E2%2B80x Start with the given expression


2x%28x%5E2-13x%2B40%29 Factor out the GCF 2x


Now let's focus on the inner expression x%5E2-13x%2B40




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Looking at 1x%5E2-13x%2B40 we can see that the first term is 1x%5E2 and the last term is 40 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 1x%5E2-13x%2B40, replace -13x with -5x%2B-8x (notice -5x%2B-8x adds up to -13x. So it is equivalent to -13x)

1x%5E2%2Bhighlight%28-5x%2B-8x%29%2B40


Now let's factor 1x%5E2-5x-8x%2B40 by grouping:


%281x%5E2-5x%29%2B%28-8x%2B40%29 Group like terms


x%28x-5%29-8%28x-5%29 Factor out the GCF of x out of the first group. Factor out the GCF of -8 out of the second group


%28x-8%29%28x-5%29 Since we have a common term of x-5, we can combine like terms

So 1x%5E2-5x-8x%2B40 factors to %28x-8%29%28x-5%29


So this also means that 1x%5E2-13x%2B40 factors to %28x-8%29%28x-5%29 (since 1x%5E2-13x%2B40 is equivalent to 1x%5E2-5x-8x%2B40)



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So our expression goes from 2x%28x%5E2-13x%2B40%29 and factors further to 2x%28x-8%29%28x-5%29


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Answer:

So 2x%5E3-26x%5E2%2B80x completely factors to 2x%28x-8%29%28x-5%29