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# The-Binomial-Theorem

last edited by 11 years, 11 months ago 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:

(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