The Binomial Theorem provides a formula for expanding the power of an expression containing exactly how many distinct terms?
1.One term, which is a monomial.
2.Three terms, which is known as a trinomial.
3.Four or more terms, known as a multinomial.
4.Two terms, which is known as a binomial.
Frequently Asked Questions
The constant term in a binomial expansion is the term that does not contain any variables (e.g., x or y). It is obtained when all the variables in the expansion are raised to the power of zero.
The numerically greatest term in a binomial expansion is the term with the highest absolute value.
No, the binomial theorem specifically applies to binomial expressions, which consist of two terms raised to a power. However, similar techniques can be used to expand expressions with more than two terms, such as trinomials or multinomials.
The Binomial Theorem is an essential tool for simplifying long expressions that follow the pattern (a + b)n. By providing a concise formula for expanding these expressions, it makes calculations more efficient. Mastery of this theorem streamlines calculations and facilitates diverse applications across mathematics, from combinatorics to probability theory.
Binomial Theorem Notes by ALLEN’s Top Maths Faculty –
1.0Binomial Theorem Statement
The Binomial Theorem is a mathematical theorem that provides a formula for expanding the powers of binomials. It states that for any real numbers a and b and any non-negative integer n, the expansion of (a+b)n is given by:
(a+b)n=∑k=0nnCkan−kbk
(a+b)n=∑k=0nCnkxnn−kyk
Where nCk represents the binomial coefficient, and equals k!(n−k)!n!
2.0Define Binomial Theorem
The Binomial Theorem refers to the mathematical principle that allows for the expansion of any positive integral power of a binomial expression into a series format. (This theorem was given by Newton)
Pascal’s Triangle
Pascal’s Triangle is a visual representation used to find binomial coefficients. It provides a convenient method for expressing the expansion of the coefficients of binomial expansions
3.0Binomial Theorem Equation
The binomial theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form (a + b)n, where "a" and "b" are any numbers or variables, and "n" is a non-negative integer. The binomial theorem equation is typically written as:
(a+b)n=∑r=0nnCran−rbr
This equation expresses the expansion of the binomial expression (a +b)n into a sum of terms, where nCrrepresents the binomial coefficient "n choose r."
Related Video:
4.0Binomial Coefficients Expansion
(a + b)n = nC0 an + nC1 an-1 b + nC2an–2b2 + … + nCr an-r br + …+ nCn-1 a1bn–1 + nCn bn
= ∑r=0nnCran−rbr
Some more important formulas of binomial theorem are:
=(n + 1) terms + n terms + (n – 1) terms + … + 1 term
Total number of terms = (n +1) + n + (n – 1) + … + 1
=(n +1)(n +2)/2
9.0Binomial Theorem JEE Mains Questions
The (n+1)th term from the end in the expansion of (2x − 1/x )3n is :
a. 3n!n!2n!⋅2n⋅x−n
b. 3n!3n!n!⋅2n⋅x−n
c. 2n!n!3n!⋅2n⋅x−n
d. 2n!3n!⋅2n⋅x−n
Solution: The correct answer is C :
2n!n!3n!⋅2n⋅x−n
(n + 1)th term from the end in the expansion of (2x − 1/x )3n is T2n + 1 from the beginning.
10.0Solved Question on Binomial Theorem
What is the binomial theorem?
Ans: - The binomial theorem is a mathematical theorem that provides a formula for expanding the powers of binomials. It states that for any real numbers a and b and any non-negative integer n, the expansion of (a+b)n is given by:
