Expert: Steve Holleran - 9/15/2008

Question
expand following questions using binomial theorem. a)(x-y)^3, b)(2x-y)^6 and c)(a-b^2)^5.please i need detail.thank you very much

Answer
Hi Ahmed,

The easiest way is to use Pascal's triangle:

                                  1
                              1       1
                           1      2       1
                         1     3      3      1
                      1      4     6     4      1
                    1    5      10    10    5     1
                  1   6      15     20    15   6    1

These give the coefficients it the binomial expansions.

For your problems, it goes like

A)  (x - y)^3  = 1*x^3*(-y)^0 + 3*x^2*(-y) ^1 + 3x^1 *(- y)^2 + 1(-y)^3

              =  x^3 - 3x^2 y + 3xy^2 - y^3

Notice that the powers of x start at 3 and go down one in each term, and the powers of y start at 0 and come up one in each term.  And, in each term, the powers add to 3.

Same for the others:

B)  (2x-y)^6 = 1(2x)^6(-y)^0  + 6(2x)^5 (-y)^1 + 15(2x)^4(-y)^2 +
  
            20(2x)^3(-y)^3 + 15(2x)^2(-y)^4 + 6(2x)^1(-y)^5 + (-y)^6


I'll leave it for you to do the multiplying and simplifying.

C)(a-b^2)^5 = (a)^5 + 5(a)^4 (-b^2)^1 + 10(a)^3 b^2)^2  + 10(a)^2     (-b^2)^3 + 5(a)^1 (-b^2)^4 + (-b^2)^5


To describe the method using combinations takes way too much time.
I hope this helps you out.
Steve