Discovered the Binomial Theorem
The-Binomial-Theorem A binomial is a polynomial with two terms.
There are several things that you must notice while looking at the expansion
- There are n+1 terms in the expansion of (x+y)n
- The degree of each term is n
- The powers on x begin with n and decrease to 0
- The powers on y begin with 0 and increase to n
- The coefficients are symmetric
For Example:
(x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
Use Pascal's Triangle to find the coefficients for the expansion:
Each row of pascals triangle gives the binomial coefficients. For example the row 1 2 1 are the coefficients of (a + b)². The next row, 1 3 3 1, are the coefficients of (a + b)3; and so on.
OR
you can use combinations to find the coefficients in a binomial:
Combinations are the nCr button on your calculator. plug the biggest number in for n and and plug how many you are selecting from for r.
for example :
(x+y)5 = x5 + (5C1)x4y + (5C2)x3y2 +(5C3)x2y3 + (5C4)xy4 + (5C5)y5
the blue is what you put into your calculator using the nCr application on your calculator.
more Examples:
www.purplemath.com/modules/binomial.htm
(x+y)0 = 1
(x+y)1 = x + y
(x+y)2 = x2 + 2xy + y2
(x+y)3 = x3 + 3x2y + 3xy2 + y3
(x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x+y)5 = x5 + 5x4y + 10x3y2 +10x2y3 + 5xy4 + y5
these links may help you better understand: The Binomial Theorem , and more help
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