SOLUTION: Could you please help me with the binomial theorem step by step. I have no idea how to this. (a-2b)to the 15 power. Thank you!

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Question 1059424: Could you please help me with the binomial theorem step by step. I have no idea how to this. (a-2b)to the 15 power. Thank you!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Look at the general binomial expansion,
%28x%2By%29%5E15=sum%28C%2815%2Cx%29%2Ax%5E%2815-n%29%2Ay%5En%2Cn=1%2C15%29
1st term n=0
C%2815%2C0%29%2Ax%5E%2815-0%29%2Ay%5E0=1%2Ax%5E15%2A1=x%5E15
.
.
2nd term n=2
C%2815%2C1%29%2Ax%5E%2815-1%29%2Ay%5E1=15%2Ax%5E14%2Ay=15x%5E14y
.
.
3rd term n=3
C%2815%2C2%29%2Ax%5E%2815-2%29%2Ay%5E2=105%2Ax%5E13%2Ay%5E2
.
.
Remaining terms,
455%2Ax%5E12%2Ay%5E3
1365%2Ax%5E11%2Ay%5E4
3003%2Ax%5E10%2Ay%5E5
5005%2Ax%5E9%2Ay%5E6
6435%2Ax%5E8%2Ay%5E7
6435%2Ax%5E7%2Ay%5E8
5005%2Ax%5E6%2Ay%5E9
3003%2Ax%5E5%2Ay%5E10
1365%2Ax%5E4%2Ay%5E11
455%2Ax%5E3%2Ay%5E12
105%2Ax%5E2%2Ay%5E13
15%2Ax%2Ay%5E14
y%5E15
Now substitute,
x=a and y=-2b
.
.
x%5E15=a%5E15
15x%5E14y=15a%5E14%28-2b%29=-30a%5E14b
105%2Ax%5E13%2Ay%5E2=105%2Aa%5E13%2A+%28-2b%29+%5E2=420a%5E13b%29=%5E2
455%2Ax%5E12%2Ay%5E3=455%2Aa%5E12%2A%28-2b%29%5E3=++++-3640a%5E12b%5E3
1365%2Ax%5E11%2Ay%5E4=1365%2Aa%5E11%2A%28-2b%29%5E4=++21840a%5E11b%5E4
3003%2Ax%5E10%2Ay%5E5=3003%2Aa%5E10%2A%28-2b%29%5E5=++-96096a%5E10b%5E5
5005%2Ax%5E9%2Ay%5E6=5005%2Aa%5E9%2A%28-2b%29%5E6=++++320320a%5E9b%5E6
6435%2Ax%5E8%2Ay%5E7=6435%2Aa%5E8%2A%28-2b%29%5E7=++++-823680a%5E8b%5E7
6435%2Ax%5E7%2Ay%5E8=6435%2Aa%5E7%2A%28-2b%29%5E8=++++1647360a%5E7b%5E8
5005%2Ax%5E6%2Ay%5E9=5005%2Aa%5E6%2A%28-2b%29%5E9=++++-2562560a%5E6b%5E9
3003%2Ax%5E5%2Ay%5E1=3003%2Aa%5E5%2A%28-2b%29%5E1=++++3075072a%5E5b%5E10
1365%2Ax%5E4%2Ay%5E11=1365%2Aa%5E4%2A%28-2b%29%5E11=++-2795520a%5E4b%5E11
455%2Ax%5E3%2Ay%5E12=455%2Aa%5E3%2A%28-2b%29%5E12=++++1863680a%5E3b%5E12
105%2Ax%5E2%2Ay%5E13=105%2Aa%5E2%2A%28-2b%29%5E13=++++-860160a%5E2b%5E13
15%2Ax%2Ay%5E14=15%2Aa%2A%28-2b%29%5E14=++++++++++245760a%5E1b%5E14
y%5E15=%28-2b%29%5E15=-32768b%5E15
So then just sum all the terms to get the binomial expansion.