Binomial Theorem for Positive Integral Indices

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

1. 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.

2. 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.

3. Within Weather Forecast Services.

4. The process of evaluating candidates.

5. Architecture, estimating cost in engineering projects.

The Binomial Theorem has many important topics.

1. A Binomial Theorem to prove Positive Integral Index

2. Pascal's Triangle

3. General The term

4. Middle In the Long

5. 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)a^kb + (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 ⁿC_r   =\[\frac{n!}{r!(n−r)!}\], where n = a non - negative integer and 0 ≤ r ≤ n.

Also , the value of ⁿC_n and ⁿC₀ is equal to 1.

The Pascal’s Triangle can be now re-written,

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