Binomial Theorem For Rational Index

The importance of binomial theorem for rational index meaning describes the algebraic expansion of powers of a binomial in elementary mathematics. The theorem states that the polynomial (x + y)n may be expanded into a sum comprising terms of the type axbyc where the exponents b and c are nonnegative integers with b + c = n and the coefficient an of each term is a specific positive integer dependent on n and b.
For instance, in the case of n = 4,

Example of the binomial theorem on a rational index

A binomial theorem for the rational index is a two-term algebraic expression. As an example, a + b, x – y, etc are binomials. When a binomial is raised to exponents, we have a set of algebraic identities to find the expansion. 2 and 3. For example, (a + b)2 = a2 + 2ab + b2.

Binomial theorem properties.

The total number of terms included in the expansion of (x + a)n + (x−a)n are (n+2)/2 if “n” is even or (n+1)/2 if “n” is odd. The total number of terms included in the expansion of (x + a)n − (x−a)n are (n/2) if “n” is even or (n+1)/2 if “n” is odd

Expression of a Binomial

A Binomial Expression is an algebraic expression with only two terms separated by the operators +, -.

Example:

1. a) x + 2y    
2. b) 3x – 5y
3. c) (a + x)  

The Positive Integral Index Binomial Theorem

The Binomial theorem for the rational index is a formula that can be used to expand any power of a binomial statement into a series. Sir Isaac Newton proved this theorem.

The binomial theorem is a rule that can be used to enlarge any power of a binomial. If n is a positive integer and x, y ∈ C are positive integers, then

(x + y)n= nC0 xny0 + nC1  xn-1y1 + nC2  xn-2y2 +…+ nCn  x0yn 

i.e., (x + y)n = r=0n nCrxn-ryr

Here  nC0,  nC1,  nC2,……  nCn are called binomial coefficients

   nC0,= n!/r!(n–r)! for 0 ≤ r ≤ n.

Binomial theorem general formula

To expand any binomial power in the form of a series, the formula of binomial theorem is used. It is as follows:

(a+b)n = r=0nnCran-rbr

Here n is considered a positive integer and a, b are real numbers and 0 < r ≤ n. 

The binomial theorem applies to negative numbers.

The generalisation of the binomial theorem for positive integer exponents to negative integer exponents is possible. This gives rise to several well-known Maclaurin series, which have a wide range of applications in calculus and other fields of mathematics.

In binomial expansion, there is a rational term.

The number of rational terms in a formula (a1/l + b1/k )n is [n / LCM of {l,k}] when neither of and is a factor of and when one of and is a factor of [n / LCM of {l,k}] + 1 where [.] is the most significant integer function

The binomial theorem was discovered by Newton.

The method was employed by Newton himself to find areas under curves. He observed that some patterns concealed in the integer binomial sequence surfaced about curves, so he extended them to rationals and came up with the generalised binomial sequence and the generalised binomial theorem.

For non-integers, the binomial theorem is applicable.

I finally found out how to use the derivative quotient to differentiate xn and get nxn-1, but it required binomial expansion for non-integer values. The first, second, last, and second-to-last terms are the most I can discover with binomial expansion.

Binomial coefficients sum

Placing x = 1 in the process of expansion (1+x)n = nC0 + nC1 x + nC2 x2 +…+ nCx xn, we get,

2n = nC0 + nC1 x + nC2 +…+ nCn.

We kept x = 1, and achieved the intended outcome i.e. ∑nr=0  Cr = 2n.

Binomial Theorem and Mathematical Reasoning removed from the CBSE syllabus

Many questions based on mixed ideas were asked in JEE Main from the deleted syllabus in previous years. Even though the Binomial Theorem and Mathematical Reasoning have been removed from the CBSE syllabus, at least one question from these chapters has been asked in the JEE Main Exam in previous years.

Conclusion

When there are just two possible outcomes for a random variable, such as success or failure, the binomial theorem for rational index distribution is employed. Both success and failure are mutually exclusive; they cannot occur simultaneously. A finite number of trials, n, is assumed in the binomial distribution. Each trial is distinct from the previous one. This means that the chance of success, p, remains constant from trial to trial. The likelihood of failure, q, is equal to 1 – p; thus, success and failure probabilities are complementary.