Introduction to Binomial Theorem
Binomial is an algebraic expression that contains two distinct terms joined by an + or - sign. Let's take a look at the following example to comprehend the distinction between monomial, trinomial, and binomial.
xy2 (Monomial term)
x-y, y+4 (Binomial term)
x2+y+1 (Trinomial term)h
The Binomial Theorem is different from a Binomial Distribution
The Binomial Theorem is a fast method to expand a binomial expression by (that is, raised to) huge powers. This is a significant topic(section) within algebra. It is applicable to Permutations and Combinations, Probability Matrices, Permutations, and Mathematical Induction. If you're studying for a competitive exam for university admissions or job applications, then this theorem will be essential for you because it is an essential and essential part of algebra. The chapter will teach you'll discover a shortcut that will enable you to calculate (x + y)n without having to multiply the binomial on its own the number of times.
A polynomial can be described as an algebraic expression composed by two or more words which are subtracted or added to, or multiplied. It can also contain variables, coefficients, exponents, constants, constants and operators like subtraction and addition. There are three kinds of polynomials: trinomial, binomial, and monomial.
Monomials are algebraic terms that have only one term while trinomials are expressions with precisely three terms.
The evidence of the binomial theorem has been known to human beings as early as in the fourth century BC. The binomial for cubes was employed during the 6th century AD. An Indian mathematician called Halayudha explained this technique using Pascal's triangular in the 10th century AD.
The precise formulation of this theory was made during the 12th century. Mathematicians applied these results into the next steps until Sir Isaac Newton generalized the binomial theorem to all exponents in 1665.
How do you Apply the Binomial Theorem?
There are some things to keep in mind when using the Binomial Theorem.
They include:
The exponents of the initial word (a) reduces the number n to zero.
Exponents for the term (b) goes from zero to N.
Exponent a is the sum of its exponents. as well as B is the same as.
Its coefficients for the first and the last term are both 1.
Application in Real-World Situations of The Binomial Theorem
The binomial theorem is utilized extensively throughout Statistical and Probability Analyses. It's extremely useful since our economy relies on Statistics as well as Probability Analysis.
In the higher mathematics and calculation field The Binomial Theorem is used in formulating equations' roots that are higher power. It is also used to prove many of the most important mathematical and physical equations.
Within Weather Forecast Services.
The process of evaluating candidates.
Architecture, estimating cost in engineering projects.
The Binomial Theorem has many important topics.
A Binomial Theorem to prove Positive Integral Index
Pascal's Triangle
General The term
Middle In the Long
Properties and Applications of the Binomial Theorem
Many interesting Properties of the Binomial Theorem
The sum of every and every word within the expanded (x+y)n is 1+n.
The sum of the indices of the two variables in each word is n .
The above expansion can also be observed when the numbers x and y are complicated numbers.
It is the coefficient that all words are equal (equal in distance to each other) starting from until the point at which they reach their conclusion.
The binomial coefficients slowly increase to their highest value, then gradually diminish.
Before knowing about Binomial theorem let’s know about expansions,
We all know about these four important expansions: (l+ m)2, (l-m)2, (l + m)3, (l – m)3
(l +m)2 = l2 + m2 + 2lm
(l-m)2 = l2 + m2– 2lm
(l + m)3 = l3 + m3 + 3lm(l + m)
(l – m)3 = l3 – m3 – 3lm(l – m)
In the above expansions,
We see that the total number of terms in the expansion is one more than the index. For example, in the expansion of (l+m)2, the number of terms is equal to 3 whereas the index of (l+ m)2 is equal to 2.
Powers of the first quantity ‘l’ generally go on decreasing by 1 whereas the powers of the second quantity which is ‘m’ increase by 1, in the successive terms.
In each term of the above expansion, the sum of the indices of l and m is the same and is equal to the index of l+m.
Using the above expansions, we can easily find out the values of,
(102)2 = (100 + 2)2
=1002 + 2 × 100 × 2 + 22
=10000 + 400 + 4 = 10404
Similarly,
(106)2 = (100 + 6)2
=1002+6 × 100 × 2 + 62
=10000 + 1200 + 36 = 11236
But, finding out the values of (103)6, (119)5 with repeated multiplication is difficult. This is the reason why we use binomial Theorem, it makes repeated multiplication easy.
What is the statement of Binomial Theorem for Positive Integral Indices -
The Binomial theorem states that “the total number of terms in an expansion is always one more than the index.”
For example, let us take an expansion of (a + b)n, the number of terms for the expansion is n+1 whereas the index of expression (a + b)n is n, where n is any positive integer.
By using the Binomial theorem, we can expand (x +y)n, where n is equal to any rational number. Let’s discuss the binomial theorem for positive integral indices.
Now, let us write the expansion of (x+y)n , 0≤n≤5,wherenisaninteger and let’s find the properties of binomial expansion:
(m+n)0=1
(m+n)1=(m+n)
(m+n)2=m2+2mn+n2
(m+n)3=m3+3m2 n+3mn2+ n3
(m+n)4=m4+4m3 n+6m 2n2+4mn3+n4
(m+n) 5=m5+5m4 n+10m3 n2+10m2 n3+5mn4+n5
What is the Binomial Theorem for Positive Integral Indices?
Binomial Theorem for index as positive integer n is denoted by :
We know that nCr = \[\frac{n!}{r!(n−r)!}\], where n = a non-negative integer and 0≤r≤n. Also , the value of nCn and nC0 is equal to 1.
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Proof of Binomial Theorem -
Using mathematical induction, let us prove binomial theorem:
P(n): (a + b)n = nC0 an + nC1 an-1 b + nC2 an-2 b2 + .........nCn-1 a bn-1 + nCn bn
For n equal to 1,
P(1): (a+ b)1 = 1C0 a1 + 1C1 b1 = a + b, which is true.
Let us suppose that P(k) is true for any positive integer k.
Then, we get,
P(k): (a+b)k = kC0ak + kC1ak-1b + kC2ak-2b2 + .........kCk-1abk-1 + kCkbk............... Equation (1)
P(k+1) : (a+b)k+1 = (k+1)C0ak+1 + (k+1)C1akb + (k+1)C2ak-1b2 + ......... + (k+1)Ck+1bk+1
(a+b)k+1
(a+b)k+1 = (a+b) (a+b)k
= (a+b) (kC0ak + kC1ak-1b + kC2ak-2b2 + .........kCk-1abk-1 + kCkbk)
= kC0ak+1 + kC1akb + kC2ak-1b2 + .........kCk-1a2bk-1 + kCkabk + kC0akb + kC1ak-1b2 + kC2ak-2b3 + ......... kCk-1abk + kCkbk+1
Grouping the like terms, we get
= kC0ak+1 + (kC1 + kC0)akb + (kC2 + kC1 )ak-1b2 + .................. (kC1 +kCk-1 ) abk + kCkbk+1
kC0 =1 = k+1C0; kCr + kCr-1 = k+1Cr, and kCk = k+1 Ck+1 = 1
⇒ (a+b)k+1 = k+1C0ak+1 + (k+1)C1akb + (k+1)C2ak-1b2 + .................. (k+1)Ckabk + (k+1)Ck+1bk+1
P (k+1) is true whenever P(k) is true.
Therefore, P(n) is true for all positive integral values of n.
Formula for Pascal's Triangle -
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Pascal’s triangle can be defined as a triangular array of binomial coefficients. The structure of the Pascal triangle was developed by a French mathematician, Blaise Pascal.
In the above diagram we see that a particular pattern is followed.
The diagram below shows how we get the values, it shows the flow of how the numbers are added in a Pascal’s triangle.
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We notice that in the triangle 3 = 1 + 2, 6 = 3 + 3 and so on.
In a Pascal triangle the terms in each row (n) generally represent the binomial coefficient for the index = n − 1 , where n = row
For example, Let us take the value of n = 5, then the binomial coefficients are 1 ,5,10, 10, 5 , 1.
We know that nCr =\[\frac{n!}{r!(n−r)!}\], where n = a non - negative integer and 0 ≤ r ≤ n.
Also , the value of nCn and nC0 is equal to 1.
The Pascal’s Triangle can be now re-written,
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FAQs on Binomial Theorem for Positive Integral Indices
1. How to study Binomial Theorem?
In the beginning, you must be aware of the Binomial Theorem. It is important to identify the general most important term in the binomial formula. Once you have a solid understanding of the theorem you will be able to begin the application.
If you're preparing for exams, try to tackle as many issues as you are able to. Don't jump to the solution immediately. Keep in mind that if your basic knowledge is well-defined, you should be able to answer every question related to this topic.
2.Where can Binomials be utilized in the real world?
A variety of examples of binomial distributions can be observed in actual life. For example, when an innovative drug is developed to treat a disease or condition, it either is able to cure the disease (it's effective) or doesn't treat the illness (it's not successful). If you buy the lottery ticket, you're going to win cash or you're not.
3. What is the binomial theorem that is employed in forecasting weather?
The binomial theorem is a method to extend an expression that is binomial but raised by any number of finite powers. The binomial Theorem is used to forecast weather conditions as well as predict the economic growth of the country over the next few years and the distribution of IP addresses.
4. Why do we require an expansion of binomials?
The theory of binomials (or binomial expansion) is the outcome of expanding the power of binomials or sums two terms. The theorem and its generalizations may be used to demonstrate results and to solve problems in algebra, combinatorics as well as calculus and other mathematical areas.
5. What is meant by binomial math theorem and What is meant by n and r in a binomial theorem?
The Binomial theorem in Mathematics states that for any given positive integer n, the expression of the nth power of the sum of any two numbers a and b may take place as the sum of n + 1 terms of the particular form. In the theorem, n – r + 1, where n is known to be equal to the exponent of any given binomial while the symbol r represents the term number.
