Find the 10th term of (x^2 y - xy^2)

Question

step by step explanation please :)

Answer 1

to have a 10th term of a binomial expansion, you need an exponent, call it "n"

n has to be greater than 8. If n=9 then the 10th term would be the last term = (-xy^2)^9 = -x^9y^18

the 9th term would be 9(x^2y)(-xy^2)^8 = 9x^3y^17. To get coefficients of other terms in the expansion, you can find them in Pascal's triangle, the line starting with 1, 9, ... then have each term decrease in power of the first term of the binomial by 1, and increase by one for the power of the 2nd term in the binomial.

It follows the same pattern as expanding (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, where n=3, the last term is n+1 = 4th term

In the given problem a=x^2y and b = -xy^2 and n>8 , if n=9 then the last term is 9+1 or the 10th term

the coefficients of each term can be found by going to Pascal's Triangle at the line beginning with 1,9... or in general, 1,n...

another method to get the coefficients is calculate the combination of 9 taken 0 at a time through 9 taken 9 at a time. 9C0 = 1, 9C1 = 9, 9C2 = 9!/7!2! = 9(8)/2 = 36, 9C3 .....9C8 =9 9C9=1