Applications of Binomial Expression

 

The binomial theorem is an important topic of the IIT JEE Mathematics. The binomial expansion has got immense applications and is extremely useful in simplifying various lengthy computations.

The binomial expansion for the nth degree polynomial is given by:

 

If we need to compute (1+x)¹⁴ that does not mean that we will have to multiply the term 14 times but rather with the help of binomial theorem it can be computed within seconds.   

We Discuss Some of the Major Applications of the Binomial Expression:

1. Divisibility problems:

Let (1 + x)ⁿ = 1 + ⁿC₁x + ⁿC₂x² +...+ ⁿC_nxⁿ.

In any divisibility problem, we have to identify x and n. The number by which division is to be made can be, x, x² or x³, but the number in the base is always expressed in form of (1+x). This is the most convenient and widely used form of binomial expression.

View the video on binomial theorem

Illustration:

Find the remainder when 7¹⁰³ is divided by 24.

Solution: Such types of questions are generally asked in the exam and are a bit tricky.

We are required to find the remainder when 7¹⁰³ is divided by 24.

Firstly, we know that 48 is divisible by 24 which can also be written as 49 is congruent to 1 mod 24. Thus, this can be simplified with the binomial expressions as

7¹⁰³ = 7(50 - 1)⁵¹ = 7(50⁵¹ - ⁵¹C₁ 50⁵⁰ + ⁵¹C₂ 50⁴⁹ - ... - 1)

        = 7(50⁵¹ - ⁵¹C₁50⁵⁰ +...+ ⁵¹C₅₀50) - 7 - 18 + 18

        = 7(50⁵¹ - ⁵¹C₁50⁵⁰ +...+ ⁵¹C₅₀50) - 25 + 18

        => remainder is 18.

Illustration:

If I is the integral part and f is the fractional part of (2 + √3)ⁿ then prove that (I + f)(I - f) = 1. Also prove that I is an odd integer.

Solution: Given expression is (2 + √3)ⁿ.

I + f where I is an integer and 0, f < 1. We have to show that I is odd and that (I + f) (I - f) = 1

Here note that (2 + √3)ⁿ (2 - √3)ⁿ = (4 - 3)ⁿ = 1

.·.(2 + √3)ⁿ(2 - √3)ⁿ =1. It is thus required to prove that (2 - √3)ⁿ = I-f

But, (2 + √3)ⁿ + (2 + √3)ⁿ = [2ⁿ - C₁.2^n-1.√3 + C₂2^n-2.(√3)² - ...]

+ [2ⁿ - C₁.2^n-1.√3 + C₂2^n-2.(√3)² - ...]

        =2[2ⁿ - C₂.2^n-2.3 + C₄2^n-4.3² - ...] = even integer

        .·.  Now 0 < (2 - √3) < 1

        .·.  0 < (2 - √3)ⁿ < 1

        .·.  If (2 - √3)ⁿ = f' then I + f + f' = Even

        Now 0 < f < 1 and 0 < f' < 1                        

        .·.  f + f' = integer

The above equations imply that f + f' = 1 (·.· 0 < f + f' < 2)

        .·.  I is odd and f' = I - f

        => (I + f)(I - f) = 1.
 

Binomial Expression with Non- Positive Exponent

The characteristics discussed hitherto were confined to positive integer n. If n takes any other value, then binomial theorem is written as:

Say, we have to find out (x+a)ⁿ n ∉ I⁺

If a > 1 then it is written as aⁿ (a + (x/a))ⁿ. This is expanded as

aⁿ (1+(x/n))ⁿ = aⁿ[1+n(x/a) + (n(n-1)/2) (x/a)² +  (n(n-1)(n-2))/3! (x/a)³ + ......∞]

Since n is not positive integer therefore the series on the right hand side will converge only for |x/a| < 1. Moreover, there are infinite terms in the expansion contrary to the binomial expansion for a positive integer n.

Remarks:

• It is important to note here that in this case it is not possible to use the symbol ⁿC_r to denote the general term.

• The expansions in ascending powers of x are valid only if x is small. In case x is large, i.e. |x| > 1, then it is more convenient to expand it in powers of 1/x.

• The following expansions must be remembered as they help in solving questions easily (for |x| <1)

(1 + x)^-1 = 1 - x + x² – x³ + x⁵ - ….. ∞

(1 - x)^-1 = 1 + x + x² + x³ + x⁴ + ….. ∞

(1 + x)^-2 = 1 - 2x + 3x² – 4x³ + ….. ∞

(1 - x)^-2 = 1 + 2x + 3x² + 4x³ + ….. ∞

Illustration:  Given positive integers r > 1, n > 2 and the coefficient of (3r)th and (r + 2)th terms in the binomial expansion of (1+x)²ⁿ are equal. Then what is the relation between n and r?

Solution: In the expansion of (1+x)²ⁿ we have

t_3r = ²ⁿC_3r-1(x)^3r-1

t_r+2 = ²ⁿC_r+1(x)^r+1

Since the binomial coefficients of t_3r and t_r+2 are equal, so

²ⁿC_3r-1 = ²ⁿC_r+1

This gives 3r -1 = r + 1 or 2n = (3r -1 + r + 1)

Hence, 2r = 2 or 2n = 4r

Hence, r = 1 or n = 2r

But r >1, so we take the value of n = 2r.

Multinomial Expression:

The generalized form of binomial expansion is called the multinomial expansion. When there are more than two terms the case is considered to be of multinomial expansion.

If n ∈ N, then the general term of the multinomial expansion (x¹ + x² + x³ +...+ x^k)ⁿ is (n!/a₁!a₂!...a_k!) (x₁^a1,x₂^a2,..…x_k^ak ), where a₁ + a₂ + a₃ +......+ a_k = n and a < a_i < n, i = 1, 2, 3, ...... k and the total number of terms in the expansion is ^n+k-1C_n-1.

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