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} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}

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} = x^{5} + (5C1)x^{4}y + (5C2)x^{3}y^{2} +(5C3)x^{2}y^{3} + (5C4)xy^{4} + (5C5)y^{5}

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} = x^{2} + 2xy + y^{2}

(x+y)^{3} = x^{3} + 3x^{2}y + 3xy^{2} + y^{3}

(x+y)^{4} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}

(x+y)^{5} = x^{5} + 5x^{4}y + 10x^{3}y^{2} +10x^{2}y^{3} + 5xy^{4} + y^{5}

these links may help you better understand: The Binomial Theorem , and more help

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