SOLUTION: I need to understand how to do this problem by factoring completely. I believe my first step is finding the common factors. Can someone show me step by step? 27y^5 - 63y^4 + 30y

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I need to understand how to do this problem by factoring completely. I believe my first step is finding the common factors. Can someone show me step by step? 27y^5 - 63y^4 + 30y      Log On


   

Question 221874: I need to understand how to do this problem by factoring completely. I believe my first step is finding the common factors. Can someone show me step by step?
27y^5 - 63y^4 + 30y^3
and
125-8u^3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1


27y%5E5-63y%5E4%2B30y%5E3 Start with the given expression


3y%5E3%289y%5E2-21y%2B10%29 Factor out the GCF 3y%5E3


Now let's focus on the inner expression 9y%5E2-21y%2B10




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Looking at 9y%5E2-21y%2B10 we can see that the first term is 9y%5E2 and the last term is 10 where the coefficients are 9 and 10 respectively.

Now multiply the first coefficient 9 and the last coefficient 10 to get 90. Now what two numbers multiply to 90 and add to the middle coefficient -21? Let's list all of the factors of 90:



Factors of 90:
1,2,3,5,6,9,10,15,18,30,45,90

-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 90
1*90
2*45
3*30
5*18
6*15
9*10
(-1)*(-90)
(-2)*(-45)
(-3)*(-30)
(-5)*(-18)
(-6)*(-15)
(-9)*(-10)

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


Now which of these pairs add to -21? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -21

First NumberSecond NumberSum
1901+90=91
2452+45=47
3303+30=33
5185+18=23
6156+15=21
9109+10=19
-1-90-1+(-90)=-91
-2-45-2+(-45)=-47
-3-30-3+(-30)=-33
-5-18-5+(-18)=-23
-6-15-6+(-15)=-21
-9-10-9+(-10)=-19



From this list we can see that -6 and -15 add up to -21 and multiply to 90


Now looking at the expression 9y%5E2-21y%2B10, replace -21y with -6y%2B-15y (notice -6y%2B-15y adds up to -21y. So it is equivalent to -21y)

9y%5E2%2Bhighlight%28-6y%2B-15y%29%2B10


Now let's factor 9y%5E2-6y-15y%2B10 by grouping:


%289y%5E2-6y%29%2B%28-15y%2B10%29 Group like terms


3y%283y-2%29-5%283y-2%29 Factor out the GCF of 3y out of the first group. Factor out the GCF of -5 out of the second group


%283y-5%29%283y-2%29 Since we have a common term of 3y-2, we can combine like terms

So 9y%5E2-6y-15y%2B10 factors to %283y-5%29%283y-2%29


So this also means that 9y%5E2-21y%2B10 factors to %283y-5%29%283y-2%29 (since 9y%5E2-21y%2B10 is equivalent to 9y%5E2-6y-15y%2B10)



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So our expression goes from 3y%5E3%289y%5E2-21y%2B10%29 and factors further to 3y%5E3%283y-5%29%283y-2%29


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Answer:

So 27y%5E5-63y%5E4%2B30y%5E3 completely factors to 3y%5E3%283y-5%29%283y-2%29


In other words, 27y%5E5-63y%5E4%2B30y%5E3=3y%5E3%283y-5%29%283y-2%29



# 2



125-8u%5E3 Start with the given expression.


%285%29%5E3-%282u%29%5E3 Rewrite 125 as %285%29%5E3. Rewrite 8u%5E3 as %282u%29%5E3.


%285-2u%29%28%285%29%5E2%2B%285%29%282u%29%2B%282u%29%5E2%29 Now factor by using the difference of cubes formula. Remember the difference of cubes formula is A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29


%285-2u%29%2825%2B10u%2B4u%5E2%29 Multiply

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Answer:
So 125-8u%5E3 factors to %285-2u%29%2825%2B10u%2B4u%5E2%29.

In other words, 125-8u%5E3=%285-2u%29%2825%2B10u%2B4u%5E2%29