Binomial Expansion

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The binomial expansion formula/binomial theorem is used to expand expressions of the form (a+b)^n where, ‘n’ is any natural number. The word binomial signifies that there are two variables ‘a’ and ‘b’.

The well known formula, (a+b)^2 = a^2 + 2ab+ b^2 is an example of the binomial expansion when n=2.

The binomial expansion formula is given as,

(a+b)^n = \sum_{r=0}^{n} {n\choose r}a^{n-r} b^r

where, {n\choose r} = \frac{n!}{r!(n-r)!} . These are known as the binomial coefficients.

Let us try to understand the above formula via some examples.

Example 1 of binomial expansion:

Suppose we take n = 2 in the above formula. Then we get,

(a+b)^2 = \sum_{r=0}^{2} {2\choose r}a^{2-r} b^r = {2\choose 0}a^2 b^0 + {2\choose 1}a^1 b^1 + {2\choose 2}a^0 b^2 = 1a^2 +2a^1b^1 +1b^2 = a^2 + 2ab + b^2 which is the usual expansion for (a+b)^2 = a^2 + 2ab+ b^2

We calculate the binomial coefficents in the expansion using the above formula. For example, {2\choose 1} = \frac{2!}{1!(2-1)!} = \frac{2}{1 \text{ x } 1} = 2

Example 2 of binomial theorem:

Suppose we take n = 4 in the above formula. Then we get,

(a+b)^4 = \sum_{r=0}^{4} {4\choose r}a^{n-r} b^r (a+b)^4 = {4\choose 0}a^4 b^0 + {4\choose 1}a^3 b^1 + {4\choose 2}a^2 b^2 + {4\choose 3}a^1 b^3 + {4\choose 4}a^0 b^4 = a^4 + 4a^3 b + 6a^2 b^2 + 4a b^3 + b^4

We calculate the binomial coefficents in the expansion using the formula for n\choose r . For example, {4\choose 2} = \frac{4!}{2!(4-2)!} = \frac{24}{2 \text{ x } 2} = 6

Notice that in the above expansion the power of ‘a’ decreases one at a time and the power of ‘b’ increases one at a time. This holds true in general.

Simplifying Calculation of binomial coefficients:

We can simplify the calculations for the binomial coefficients using the Pascals triangle. The Pascals triangle looks like,

To construct the binomial triangle we add two consecutive numbers in the preceding row to obtain the number in the succeeding row. For example, to obtain the number 35 in the eighth row, we add the numbers 15 and 20 lying just above it in the previous row.

The Pascal triangle gives us the binomial coefficients which we can use in the binomial expansion.

Example 3 of binomial theorem:

If we want to expand (a+b)^5 we expand it using the binomial formula and use the coefficients in the sixth row of the above binomial triangle.

(a+b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5

Using the binomial formula and Pascals triangle we may similarly expand any expression of the form (a+b)^n.

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