what is the expanded form of (4x-y)^10

Question

Factoring binomials.

Answer 1

First list all the coefficients for the binomial expansion:

1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1,

using C(10,r)=10!/(r!(10-r)!).

Next we need powers of 4 from 10 to 1:

1048576, 262144, 65536, 16384, 4096, 1024, 256, 64, 16, 4.

Now we can put the expansion together:

1048576x¹⁰-10(262144x⁹y)+45(65536x⁸y²)-120(16384x⁷y³)+

210(4096x⁶y⁴)-252(1024x⁵y⁵)+210(256x⁴y⁶)-120(64x³y⁷)+

45(16x²y⁸)-10(4xy⁹)+y¹⁰.

Combining coefficients:

1048576x¹⁰-2621440x⁹y+2949120x⁸y²-1966080x⁷y³+

860160x⁶y⁴-258048x⁵y⁵+53760x⁴y⁶-7680x³y⁷+720x²y⁸-40xy⁹+y¹⁰.