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.

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) (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 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 Number	Second Number	Sum
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
-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 , 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 factors to

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