Please give the binomial theorem and explain it fully?

the binomial theorem just says how binomial expansion is done. You can use Pascal's Triangle to find the coefficients:

1

11

121

1331

1461

and so on

You do this by adding the two numbers next to each other, then writing the answer in the step below. the top step is for 0th power, second step is 1st power, third step is 2nd power, and so on. if you wanted (x+y)^3 then you'd go to the 4th step, which is 1331. so the coefficients, in order, are 1331. the exponent of the x and y go like: x3y0 x2y1 x1y2 x0y3 . so (x+y)^3 would be

x^3+3yx^2+3xy^2+y^3