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The binomial theorem states the principle for expanding the algebraic expression (x + y) n and expresses it as a sum of the terms involving individual exponents of variables x and y. Each term in a binomial expansion is associated with a numeric value which is called coefficient.
- Pascal's Triangle
A pascal's triangle is an arrangement of numbers in ...
- Principle of Mathematical Induction
Now, each step that is used to prove the theorem or ...
- Binomial Expansion Formulas
The binomial expansion formulas are used to find the ...
- Binomial Distribution
The binomial distribution represents the probability ...
- Algebraic Expression
Algebraic Expressions Algebraic expressions are the ...
- Exponents
An exponent of a number shows how many times we are ...
- Algebraic Identities
The algebra identities for three variables also have ...
- Coefficient
To find a coefficient of a variable in a term, follow ...
- Pascal's Triangle
2024年6月10日 · The binomial theorem is a formula for expanding binomial expressions of the form (x + y) n, where ‘x’ and ‘y’ are real numbers and n is a positive integer. The simplest binomial expression x + y with two unlike terms, ‘x’ and ‘y’, has its exponent 0, which gives a
There are some proofs for the general case, that $$(a+b)^n=\sum_{k=0}^n {n \choose k}a^kb^{n-k}.$$ This is the binomial theorem. One can prove it by induction on n: base: for $n=0$ , $(a+b)^0=1=\sum_{k=0}^0{n \choose k}a^kb^{n-k}={0\choose0}a^0b^0$ .
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial ( x + y ) n into a sum involving terms of the form ax b y c , where the exponents b and c are nonnegative integers with b + c = n , and the coefficient a ...
- Proof
- Generalizations
- Usage
There are a number of different ways to prove the Binomial Theorem, for example by a straightforward application of mathematical induction. The Binomial Theorem also has a nice combinatorial proof: We can write . Repeatedly using the distributive property, we see that for a term , we must choose of the terms to contribute an to the term, and then e...
The Binomial Theorem was generalized by Isaac Newton, who used an infinite series to allow for complex exponents: For any real or complex , , and ,
Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial , one could factor it as such: . It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients.
2023年10月5日 · The General Binomial Theorem was first conceived by Isaac Newton during the years $1665$ to $1667$ when he was living in his home in Woolsthorpe. He announced the result formally, in letters to Henry Oldenburg on $13$th June $1676$ and $24$th October $1676$ but did not provide a proper proof (at that time the need for the appropriate ...
The binomial theorem (or binomial expansion) is a result of expanding the powers of binomials or sums of two terms. The coefficients of the terms in the expansion are the binomial coefficients \( \binom{n}{k} \).
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