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Вы можете ознакомиться и скачать презентацию на тему Binomial theorem. Доклад-сообщение содержит 10 слайдов. Презентации для любого класса можно скачать бесплатно. Если материал и наш сайт презентаций Mypresentation Вам понравились – поделитесь им с друзьями с помощью социальных кнопок и добавьте в закладки в своем браузере.

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Binomial theorem

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Binomial theorem. This is the formula that represents the expression for a positive integer as a polynomial: Note that the sum of the exponents of and is constant and equal to .

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- Binomial coefficients, - non-negative integer Example: From the set {1,2,3,4}, select all the possible combinations of two elements, = {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} It turns out six options. Substituting values into the formula, we check the result: binomial coefficient Example

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The properties of binomial coefficients The properties of binomial coefficients 1. The sum of the coefficients of expansion is equal to. It is sufficient to put = 1. Then the right side of Newton's binomial expansion we will have sum of binomial coefficients, and on the left:

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2. The coefficients members equidistant from the ends of the expansion, are equal. 2. The coefficients members equidistant from the ends of the expansion, are equal. This property follows from the relation:

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3. The amount of the even terms in the expansion coefficient equal to the sum of the odd terms in the expansion coefficients; each of them is 3. The amount of the even terms in the expansion coefficient equal to the sum of the odd terms in the expansion coefficients; each of them is To prove this we use the binomial: Here the even members are sign, and the odd - . As a result turns decomposition 0, therefore, the amount of binomial coefficients are equal to each other, so each of them is: get to prove.

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Examples of using the binomial of Newton Examples of using the binomial of Newton Example 1. Arrange bean in powers of . We use the binomial theorem: The values of the binomial coefficients consistently find the formula For example