SOLUTION: Factor completely: 5y^9 - 53y^6 + 72y^3. I think that this problem has to do with cubing I am just not sure how to factor it.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor completely: 5y^9 - 53y^6 + 72y^3. I think that this problem has to do with cubing I am just not sure how to factor it.      Log On


   

Question 132328: Factor completely:
5y^9 - 53y^6 + 72y^3.
I think that this problem has to do with cubing I am just not sure how to factor it.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

5y%5E9-53y%5E6%2B72y%5E3 Start with the given expression


y%5E3%285y%5E6-53y%5E3%2B72%29 Factor out the GCF y%5E3


Now let's focus on the inner expression 5y%5E6-53y%5E3%2B72




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Looking at 5y%5E6-53y%5E3%2B72 we can see that the first term is 5y%5E6 and the last term is 72 where the coefficients are 5 and 72 respectively.

Now multiply the first coefficient 5 and the last coefficient 72 to get 360. Now what two numbers multiply to 360 and add to the middle coefficient -53? Let's list all of the factors of 360:



Factors of 360:
1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360

-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-18,-20,-24,-30,-36,-40,-45,-60,-72,-90,-120,-180,-360 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 360
1*360
2*180
3*120
4*90
5*72
6*60
8*45
9*40
10*36
12*30
15*24
18*20
(-1)*(-360)
(-2)*(-180)
(-3)*(-120)
(-4)*(-90)
(-5)*(-72)
(-6)*(-60)
(-8)*(-45)
(-9)*(-40)
(-10)*(-36)
(-12)*(-30)
(-15)*(-24)
(-18)*(-20)

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


Now which of these pairs add to -53? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -53

First NumberSecond NumberSum
13601+360=361
21802+180=182
31203+120=123
4904+90=94
5725+72=77
6606+60=66
8458+45=53
9409+40=49
103610+36=46
123012+30=42
152415+24=39
182018+20=38
-1-360-1+(-360)=-361
-2-180-2+(-180)=-182
-3-120-3+(-120)=-123
-4-90-4+(-90)=-94
-5-72-5+(-72)=-77
-6-60-6+(-60)=-66
-8-45-8+(-45)=-53
-9-40-9+(-40)=-49
-10-36-10+(-36)=-46
-12-30-12+(-30)=-42
-15-24-15+(-24)=-39
-18-20-18+(-20)=-38



From this list we can see that -8 and -45 add up to -53 and multiply to 360


Now looking at the expression 5y%5E6-53y%5E3%2B72, replace -53y%5E3 with -8y%5E3%2B-45y%5E3 (notice -8y%5E3%2B-45y%5E3 adds up to -53y%5E3. So it is equivalent to -53y%5E3)

5y%5E6%2Bhighlight%28-8y%5E3%2B-45y%5E3%29%2B72


Now let's factor 5y%5E6-8y%5E3-45y%5E3%2B72 by grouping:


%285y%5E6-8y%5E3%29%2B%28-45y%5E3%2B72%29 Group like terms


y%5E3%285y%5E3-8%29-9%285y%5E3-8%29 Factor out the GCF of y%5E3 out of the first group. Factor out the GCF of -9 out of the second group


%28y%5E3-9%29%285y%5E3-8%29 Since we have a common term of 5y%5E3-8, we can combine like terms

So 5y%5E6-8y%5E3-45y%5E3%2B72 factors to %28y%5E3-9%29%285y%5E3-8%29


So this also means that 5y%5E6-53y%5E3%2B72 factors to %28y%5E3-9%29%285y%5E3-8%29 (since 5y%5E6-53y%5E3%2B72 is equivalent to 5y%5E6-8y%5E3-45y%5E3%2B72)



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So our expression goes from y%5E3%285y%5E6-53y%5E3%2B72%29 and factors further to y%5E3%28y%5E3-9%29%285y%5E3-8%29


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Answer:

So 5y%5E9-53y%5E6%2B72y%5E3 factors to y%5E3%28y%5E3-9%29%285y%5E3-8%29