BINOMIAL THEOREM : AKSHAY MISHRA XI A ‚ K V 2 ‚ GWALIOR In elementary algebra‚ the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem‚ it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc‚ where the coefficient of each term is a positive integer‚ and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients.
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The binomial theorem is a simplified way of finding the expansion of a binomial to a certain power. We can of course find the expanded form of any binomial to a certain power by writing it and doing each step‚ but this process can be very time consuming when you get into let’s say a binomial to the 10th power. Example: (x+y)^0=1 of course because anything to the power if 0 equal 1 (x+y)^1= x+y anything to a power of 1 is just itself. (x+y)^2= (x+y)(x+y) NOT x^2+y^2. So expand (x+y)(x+y)=x^2+xy+yx+y^2
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BINOMIAL THEOREM OBJECTIVES Recognize patterns in binomial expansions. Evaluate a binomial coefficient. Expand a binomial raised to a power. Find a particular term in a binomial expansion Understand the principle of mathematical induction. Prove statements using mathematical induction. Definition: BINOMIAL THEOREM Patterns in Binomial Expansions A number of patterns‚ as follows‚ begin to appear when we write the binomial expansion of a b n‚ where n is a positive integer
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1 10/10/01 Fermat’s Little Theorem From the Multinomial Theorem Thomas J. Osler (osler@rowan.edu) Rowan University‚ Glassboro‚ NJ 08028 Fermat’s Little Theorem [1] states that n p −1 − 1 is divisible by p whenever p is prime and n is an integer not divisible by p. This theorem is used in many of the simpler tests for primality. The so-called multinomial theorem (described in [2]) gives the expansion of a multinomial to an integer power p > 0‚ (a1 + a2 + ⋅⋅⋅ + an ) p = p k1 k2 kn a1 a2
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Triangle The Pascal’s Triangle is a triangular array of the binomial coefficients. The system after French mathematician Blaise Pascal. The set of numbers that form Pascal’s triangle were known before Pascal. However‚ Pascal developed many uses of it and was the first one to organize all the information together in his treatise‚ Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks’ study of figurate numbers. The
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for the Binomial Distribution P(S) The symbol for the probability of success P(F) The symbol for the probability of failure p The numerical probability of a success q The numerical probability of a failure P(S) = p and P(F) = 1 - p = q n The number of trials X The number of successes The probability of a success in a binomial experiment can be computed with the following formula. Binomial Probability
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Isaac Newton Isaac Newton was born on December 25‚ 1642‚ in Woolsthorpe‚ England. His father died before he was born‚ and his mother mother‚ Hannah Newton‚ remarried and moved away. She left Newton to be raised by his uncle. In 1654‚ he was sent to the local grammar school‚ then he enrolled at Trinity College‚ at the University of Cambridge‚ in 1661. He received his bachelor of arts in 1665‚ and was named a fellow of the College two years later. In 1666‚ Newton made three of his greatest discoveries
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Lesson 1 Assignment Questions Describe the scientific system by which plants are classified‚ in a report of up to 500 words. In this report‚ Cover: *the significance of the binomial system *why common names of plants should not be used to correctly identify plants. The scientific system to classify and naming plants are controlled and coordinated by botanist throughout the world. The system of classification in plants is to classify them in groups with similar characteristics. Then
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Binomial nomenclature (also called binominal nomenclature or binary nomenclature) is a formal system of naming species of living things by giving each a name composed of two parts‚ both of which use Latin grammatical forms‚ although they can be based on words from other languages. Such a name is called a binomial name (which may be shortened to just "binomial")‚ a binomen or a scientific name; more informally it is also called a Latin name. The first part of the name identifies the genus to which
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Math Sir Isaac Newton On Christmas day in 1642 a premature baby by the name of Isaac Newton was born. Isaac was born in Woolsthorpe‚ Lincolnshire‚ England and attended day school there while living with his grandmother. When Isaac was about thirteen‚ he went away to Grammar School in Grantham where he noticed his interest in chemicals. In 1661‚ he went to Cambridge University and stayed there until 1696 (with some gaps due to the plague). Within his Cambridge years‚ Isaac
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