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BINOMIAL THEOREM OF NEWTON By: Mayra Aydarbekova. BINOMIAL THEOREM OF NEWTON By: Mayra Aydarbekova.

Binomial is a polynomial with two terms. Binomial theorem describes algebraic expansion of power Binomial is a polynomial with two terms. Binomial theorem describes algebraic expansion of power of a binomial. Example for a binomial: 6 x – 3; 2 t – 5.

History of Binomial Theorem The 4 th century B. C. – Greek mathematician Euclid History of Binomial Theorem The 4 th century B. C. – Greek mathematician Euclid The 3 rd century B. C. – Indian mathematician Pingala The 10 th century A. D. – Halayudha & Al-Karaji The 11 th century – Persian poet and mathematician Omar Khayyam The 13 th century – Chinese mathematician Yang Hui

We know that an Islamic mathematician named al-Karaji (d. 1029) constructed a table of We know that an Islamic mathematician named al-Karaji (d. 1029) constructed a table of binomial coefficients up to (a+b)5 (that is, Pascal's triangle), and later Muslim mathematicians credited him with discovering the formula for the expansion of (a + b)n. Furthermore, in a now lost work, Omar Khayyam (1048 -1131) apparently gave a method for finding nth roots based on the binomial expansion and binomial coefficients. And in Europe, already a century before Newton's birth, Blaise Pascal's Treatise on the Arithmetical Triangle provided a handy way to generate binomial coefficients. All of these methods for binomial expansion, however, work only for positive integer values of n.

What Newton discovered was a formula for (a+b)n that would work for all values What Newton discovered was a formula for (a+b)n that would work for all values of n, including fractions and negatives: (a+b)n = an + nan-1 b + [n(n-1)an-2 b 2] / 2! + [n(n 1)(n-2)an-3 b 3] / 3! +. . . + bn

All of these scientists derived same results in different centuries. Triangular arrangement of the All of these scientists derived same results in different centuries. Triangular arrangement of the binomial coefficients are attributed to Blaise Pascal.

Exponents(powers) of binomials Exponent of 0 When an exponent is 0, you get 1: Exponents(powers) of binomials Exponent of 0 When an exponent is 0, you get 1: (a+b)0 = 1 Exponent of 1 When the exponent is 1, you get the original value, unchanged: (a+b)1 = a+b Exponent of 2 An exponent of 2 means to multiply by itself: (a+b)2 = (a+b) = a 2 + 2 ab + b 2

Exponent of 3 For an exponent of 3 just multiply again: (a+b)3 = (a+b)(a Exponent of 3 For an exponent of 3 just multiply again: (a+b)3 = (a+b)(a 2 + 2 ab + b 2) = a 3 + 3 a 2 b + 3 ab 2 + b 3 Now, notice the exponents of a. They start at 3 and go down: 3, 2, 1, 0:

Likewise the exponents of b go upwards: 0, 1, 2, 3 which are called Likewise the exponents of b go upwards: 0, 1, 2, 3 which are called coefficients.

That coefficients actually make up Pascal’s Triangle. Each number is just a sum Of That coefficients actually make up Pascal’s Triangle. Each number is just a sum Of numbers above. Ex: 1+3=4.

Newton’s own words… Newton’s own words… "Plato is my friend, Aristotle is my friend, but my best friend is truth”, 1664. "I know not what I appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell, while the great ocean of truth lay all undiscovered before me. "

Thanks For Attention! Thanks For Attention!