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The binomial theorem describes how to express and evaluate a binomial’s powers. This theorem describes how to expand and express a term of the form (a+b)n in the form rasbt, where the exponents s and t are non-negative integers that satisfy the condition s + t = n. r is a positive integer coefficient. Binomial coefficients are the words involved, and because it is for positive indices, it only grows the positive powers.
Binomial theorem
Euclids, a prominent Greek mathematician, first mentioned the binomial theorem in the 4th century BC. The binomial theorem explains how to expand the algebraic statement (x + y)n into a sum of terms using individual exponents of the variables x and y. Each word in a binomial expansion is given a numerical value called coefficient.
Coefficients of Binomial Theorem
The values of the binomial coefficients follow a distinct pattern that can be seen as a Pascal’s triangle. A triangular arrangement of binomial coefficients is known as Pascal’s triangle. It is named after Blaise Pascal, a French mathematician. All of the boundary elements of the numbers in Pascal’s triangle are 1, and the remaining numbers within the triangle are set so that each number is the sum of two numbers just above it.
Binomial Theorem for Positive Integral Indices is Stated as Follows
A binomial expression is an algebraic expression of the form a+b. Although raising a+b to any power is simple in concept, doing it to a really high power could be tiresome. The phrases will then be multiplied repeatedly, which will be a tiresome operation. However, such high powers of binomial expression are employed to determine higher power roots of equations.
“In the expansion of (a+b)n, the total number of words is one more than the index, n. The number of words in the expansion of (a+b)n is n+1, where n is the index and is a positive integer.
Statement: if n is a positive integer then,
x+a)n= nC0Xna0+ nC1Xn-1a1+n2
Xn-2.a2+…..+nCn X0an
Proof:
Let p(n):(x+a)n= nC0 Xna0+ nC1
Xn-1a1+nC2Xn-2a2+…………………..+ nCn X0an
Step1: prove that P(1) is true
If when n=1, L.H.S =(x+a)1=X+a;
R.H.S = 1C0 X0an=X.+1.1.a=X+a
L.H.S = R.H.S
P(1) is true
Binomial Theorem’s Characteristics
The Binomial Theorem has the following properties are given below:
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The expansion has n+1 words, the first of which is a and the final of which is bn.
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From term to term, the exponents of a fall by 1, whereas the exponents of b grow by 1.
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In each phrase, the sum of the exponents of a and b is n.
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The terms in a binomial expansion equidistant from both ends have equal coefficients. Coefficients are symmetric, in other words.
Conclusion
In this article we conclude that the binomial theorem is helpful in performing binomial expansions and determining expansions for algebraic identities. In addition, the binomial theorem is applied to binomial expansion in probability. The Binomial Theorem explains how to expand statements with the form (a+b)n. The bigger the power, the more difficult it is to directly expand expressions like this. And the Statement of the Binomial Theorem for Positive Integral Index is “The total number of terms in the expansion is one more than the index,” the theorem states.
