The Binomial theorem in maths is a method to expand complex algebraic terms that are in an expression with another dissimilar term. For instance, a^3 + b^7. This method helps solve complex algebraic terms raised to high powers with relative ease using binomial properties and basic algebra. In Class 11 Maths Ncert Solutions Chapter 8, the major focus is provided on binomial theorem properties, which is the process of solving for coefficients for positive integers and deriving the general and middle terms of an algebraic binomial expression. 

 

A few important binomial properties include the following:

• • Total no. of terms in the expansion of (x+y)n  is equal to (n+1)

  • The sum of the exponents of x and y is always equal to ‘n’.

  • nC0, nC1, nC2, … .., nCn or C0, C1, C2, ….., Cn are known as binomial coefficients

  • The binomial coefficients which lie equidistant from the start and end are equal. This can be represented as C0 = nCn, nC1 = nCn-1 , nC2 = nCn-2 ,….. etc.

According to the Binomial theorem, the total number of terms in an expansion is always one more than the index or the sum of the exponents in the binomial expression. For instance, to expand the expression (a + b)ⁿ, the no. of terms for the expansion is n+1 whereas the index of expression  (a + b)ⁿ is n, where n is any positive integer.

 

For example if n:  0 ≤ n ≤ 5; where ‘n’ is a positive integer

The consequent binomial expansions are as follows:

(a+b)⁰= 1

(a+b)¹= (a+b)

(a+b)²= a²+2ab+b²

(a+b)³= a³+3a² b+3ab²+ b³

(a+b)⁴= a⁴+4a³ b+6a²b²+4ab³+b⁴

(a+b)⁵= a⁵+5a⁴ b+10a³ b²+10a² b³+5ab⁴+b⁵

 

General form of Binomial Theorem for positive integer indices:

In binomial theorem, the general term represents the term that is true for all the particular terms of a binomial equation. For instance, if (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn, the General Term = Tr+1 = nCr xn-r . yr. Similarly, the middle term is the term that appears at the centre of an expanded binomial equation. For instance, in (a+b)²= a²+2ab+b²   the middle term is 2ab.

1. Solve: (√2 + 1)^5 + (√2 − 1)^5

 

Solution: 

 

From the general binomial theorem expression, we can write:

(x + y)^5 + (x – y)^5 = 2[5C0  x^5 + 5C2  x^3 y^2  +  5C4 xy^4] where x = √2 and y = 1.

Thus we get,

= 2(x^5 + 10 x3y^2 + 5x y^4);

Now if we replace the values of x and y, we get:

 [(√2 + 1)^5 + (√2 − 1)^5 = 2[(√2)^5 + 10(√2)^3(1)^2 + 5(√2)(1)^4]

Which upon further solving gives us our required answer = 58√2

 

2. Find the middle term in the expansion of 2ax - (b^12/x^2)

 

Solution

 

Given binomial expression: 2ax - (b^12/x^2)

Using the properties of binomial terms, since the power to which the terms are raised is even, there is 1 middle term, which is the 7th term of the expanded binomial expression.

Furthermore, we can write:

T7 = 12C6 (2ax)^6 (-b/x^2) ^6

=> T7 = 12C6 (2^6a^6x^6) (-b)^6 / x^12

=> T7 = 12C6 (2^6a^6b^6)/x^6

 

Further putting in the value of 12C6 we arrive at the result: 59136a^6b^6 

Thus, the middle term in the expansion of 2ax - (b^12/x^2) is 59136a^6b^6 (Ans)

 

3. Show that 2^(4n+ 4) - 15n - 16 is divisible by 225. Assume that n is a Natural number.

 

Solution:

 

Given expression is: 2^(4n+ 4) - 15n - 16

Expanding this binomial expression using the properties of binomial theorem, we get:

 

= 2^4(n+ 1) - 15n - 16

= 16 ^(n+1) - 15n -16

= (1 + 15)^(n +1) - 15n -16

Now if we binominally expand this equation, we get:

[(n+1)C0 15^0 +(n+1) 1C1 15^1 + (n+1) 1C2 15^2 + (n+1) 1C3 15^3 + (n+1) 1C4 15^4 + …. + (n+1) 1C(n+1) 15^(n+1) - 15n -16 ]

 

Further simplifying we get:

 

15^2 [(n+1) 1C2 15^2 + (n+1) 1C3 15^3 + (n+1) 1C4 15^4 + ….so on]

To prove that 2^(4n+ 4) - 15n - 16 is divisible by 225, we can check if the above equation in the simplified form is divisible by 225. 

Thus, 2^(4n+ 4) - 15n - 16 is divisible by 225.

What is the concept of the Binomial Theorem?

 

In math, the binomial theorem is a method to expand algebraic expressions with finite exponents into relevant terms. This helps to find special terms of the equation in the expanded form without actually having to solve the entire complex expression by merely using the binomial properties.

What are positive integers in the Binomial Theorem?

 

Positive integer indices are the powers or exponents to which each term is raised in a binomial expression. And, positive integers are the terms by which each term of a binomial expression is multiplied.

What topics are covered in Class 11 Maths Chapter 8 Binomial theorem?

 

In Class 11 Maths Chapter 8 Binomial theorem, various sub-topics are covered. This includes properties of binomial expressions, solving binomial expressions for the middle term or a term in a specific position, solving for the independent term and the coefficient, among other such variations.