Table of contents  
Need for Binomial Theorem  
Pascal’s Triangle  
Binomial theorem for any positive integer n  
Solved Examples on Binomial Theorem 
As we examine these expansions, a few patterns emerge:
So, what's the story behind these patterns? Let's unravel the mysteries together as we explore the underlying principles of these binomial expansions.
According to the binomial theorem, it is possible to expand any nonnegative power of binomial (x + y) into a sum of the form
Binomial Theorem
The binomial theorem is expressed using the sigma notation
Where, represents the binomial coefficient, and a and b are constants. This formula expands (a+b)^{n} and is a shorthand way of writing out the terms.
The coefficients in the binomial theorem are known as binomial coefficients. These coefficients represent the number of ways to choose k elements from a set of n elements.
The expansion of (a+b)^{n} has n+1 terms, which is one more than the index n. Each term corresponds to a specific power of a and b in the binomial expression.
Example 1: Expand (x+y)^{3} using binomial theorem
Solution: The binomial theorem formula for this expansion is:
Now, let's calculate each term:
Now, combine these terms
(x + y)^{3} = x^{3} + 3x^{2}y + 3xy^{2} + y^{3}
This is the expanded form of (x+y)^{3}
Each term represents a unique combination of powers of x and y, and the coefficients come from Pascal's Triangle.
We can use a similar approach for higher powers or different binomial expressions. The binomial theorem provides a systematic way to expand such expressions without going through the cumbersome process of repeated multiplication.
Example 2: Expand (x^{2} + 2)^{6}
Solution:
(x^{2} +2)^{6} = ^{6}C_{0 }(x^{2})^{6}(2)^{0} + ^{6}C_{1}(x^{2})^{5}(2)^{1} + ^{6}C_{2}(x^{2})^{4}(2)^{2} + ^{6}C_{3 }(x^{2})^{3}(2)^{3} + ^{6}C_{4 }(x^{2})^{2}(2)^{4} + ^{6}C_{5 }(x^{2})^{1}(2)^{5} + ^{6}C_{6 }(x^{2})^{0}(2)^{6}
= (1) (x^{12}) (1) + (6) (x^{10}) (2) + (15) (x^{8}) (4) + (20) (x^{6}) (8) + (15) (x^{4}) (16) + (6) (x^{2}) (32) + (1)(1) (64)
= x^{12} + 12 x^{10} + 60 x^{8} + 160 x^{6} + 240 x^{4} + 192 x^{2} + 64
Example 3: Expand the expression (√2 + 1)^{5} + (√2 − 1)^{5 }using the Binomial formula.
Solution:
(x + y)^{5} + (x – y)^{5} = 2[5C_{0} x^{5} + 5C_{2} x^{3} y^{2} + 5C_{4} xy^{4}]
= 2(x^{5 }+ 10 x^{3} y^{2 }+ 5xy^{4})
= (√2 + 1)^{5 }+ (√2 − 1)^{5 }= 2[(√2)^{5 }+ 10(√2)^{3}(1)^{2 }+ 5(√2) (1)^{4}]
=58√2
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1. What is the significance of the Binomial Theorem in mathematics? 
2. How is Pascal’s Triangle related to the Binomial Theorem? 
3. Can the Binomial Theorem be applied to any positive integer value of n? 
4. Can you provide an example of how the Binomial Theorem is used in solving mathematical problems? 
5. What is the importance of understanding the Binomial Theorem for students preparing for exams like JEE? 
209 videos443 docs143 tests


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