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About Binomial Theorem Class 11 Chapter 8
Binomial Theorem Class 11: The binomial theorem is a theorem in mathematics that states that the sum of two terms is the product of the first term and the binomial coefficient of the second term.
Therefore the binomial theorem is usually written as:
(x + y)n = xn + yn + (xn-1y + xn-2y2 + … + yn-1)
Where:
x is the first term
n is the number of terms
y is the second term
The binomial theorem can be used to calculate the expansion of a binomial expression.
NCERT Solutions for Binomial Theorem Class 11 Maths Chapter 8
The Binomial Theorem is a mathematical theorem that describes the way in which a binomial expansion behaves. Therefore theorem is named for the mathematician and philosopher Gottfried Wilhelm Leibniz, who published it in 1676.
The theorem states that, for any real number x and any positive integer n, the following equation holds true:
(x + y)n = xn + yn + nxyn
The theorem can expanded to cover complex numbers by taking the imaginary unit i to be equal to −1. In this case, the equation becomes:
(x + y)n = xn + yn − nxyn
The binomial theorem can proven using mathematical induction.
Definition of Binomial Theorem
The binomial theorem is a mathematical theorem that states that the expansion of a binomial (that is, the sum of two terms) is a sum of terms in which each term is the product of a power of the binomial’s two factors. The theorem named for the mathematician and theologian Pierre de Fermat, who first stated it in 1654.
Binomial Expansion
In mathematics, binomial expansion is the expansion of a binomial (a sum of two terms) into a series of terms, each of which is a product of the two original terms. Therefore the expansion usually written using the symbols of mathematics, with the terms in the series arranged in descending powers of the variable x.
- For example, the expansion of (x + y)5 is
- The coefficients a k are called the binomial coefficients.
- The binomial expansion is an important technique in mathematics, and also has many applications. One of the most famous applications is the proof of the binomial theorem.
Terms in the Binomial Expansion
The binomial theorem states that for any real number x and also any positive integer n,
(x + y)n = xn + yn + (n − 1)xn−1y + (n − 2)xn−2y2 + (n − 3)xn−3y3 + ···
The coefficients of xn, yn, xn−1y, xn−2y2, and xn−3y3 in this expansion are
1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.
What is a Binomial Expression?
A binomial expression is an algebraic expression that contains two terms, separated by a plus or minus sign.
Topics and Subtopics of Binomial Theorem
Binomial theorem
-The binomial theorem states that for any real number “n” and also any positive integer “k”,
(x + y)^k = x^k + y^k + kx^k y^k
-This equation can used to calculate the value of a binomial expansion, which is an expression that contains powers of binomial.
Binomial Theorem Formula
The binomial theorem is a mathematical formula that states that a binomial expansion of a positive integer power of a binomial is the sum of all the positive integer powers of the individual binomial coefficients.
Properties of the Binomial Expansion (x + y)n
- The binomial expansion is symmetrical about the middle term.
- binomial expansion is linear.
- The binomial expansion is a polynomial.
- binomial expansion is unique.
FAQs
Q: What is the Binomial Theorem? A: The Binomial Theorem is a mathematical formula that allows us to expand expressions of the form (a+b)^n, where a and b are any two numbers and n is a positive integer.
Q: What is the formula for the Binomial Theorem? A: (a+b)^n = C(n,0)a^nb^0 + C(n,1)*a^(n-1)*b^1 + C(n,2)*a^(n-2)*b^2 + … + C(n,n)a^0b^n, where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient.
Q: What is a binomial coefficient? A: A binomial coefficient is a number that represents the number of ways to choose k items from a set of n items. The binomial coefficient is denoted by C(n,k) or (n choose k).
Q: How do I calculate binomial coefficients? A: The binomial coefficient can be calculated using the formula C(n,k) = n!/(k!(n-k)!), where n! is the factorial of n.
Q: What is a factorial? A: The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
