The term "binomial" refers to an equation that has 2 terms. An example of a binomial would be: x + y = 0 – in this example, the first term is x and the second term is y.
Binomial theorem states that when a binomial equation, like the one in the example above, is raised to a certain power (exponent), the expansion will have the following properties:
The number of terms will be equal to one greater than the value of the exponent of the binomial.
The powers of the second term in the binomial will be increasing by one in each successive term starting from zero, while the powers of the first term of the binomial will decrease by one for each successive term starting from the exponent of the binomials.
The sum of the powers in each term is equal to the exponent of the binomial.
The coefficients start from one, increases in each term, then decreases back to one. (Coefficient refers to the constant value in the term.)
To better understand these properties, let us list down some binomial expansions and examine their similarities.
(x+y)^0 = 1
(x+y)^1 = x + y
(x+y)^2 = x^2 + 2xy + y^2
(x+y)^3 = x^3 + 3(x^2)y + 3xy^2 + y^3
Let us take a closer look at each of these binomial expansions and see if they do match the properties mentioned earlier.
The first example is a binomial raised to zero. Zero plus one (0 + 1) is equal to one. The expansion has only one term. Therefore, the first property is met. The same goes with other examples.
Look at the third example. The second term of the binomial is y, as with the other examples. If you examine the exponents of y starting from the first term in the expansion (0 because y^0 is equal to 1) up to the last term (2), you will notice that the exponent increases by one in each progressive term. On the other hand, x, which is the first term, has exponents that start from two (first term) up to zero (last term). Thus, the second property holds true. And again this is the same with the other examples.
Now let's examine example number 4. In the first term, the exponent of x is 3 while y has 0 (x^3 * y^0 = x^3). The sum of their exponents is 3. In the second term, the exponent of x is 2 while y has 1 (x^3 * y^1 = (x^3)y) and the sum of the two exponents is 3. The same goes with the 2 remaining terms. This matches what was described in the third property.
Again, let's look at the fourth example. As mentioned earlier, coefficients refer to the constant values present in each term. In the fourth example, the coefficient of the first term is 1, and for the second and third term, it's 3, and for the last term is also 1. If you notice, the coefficient starts from number one, goes up to three in two terms, and goes back to one. This is the fourth property described earlier.
When you sum up all these properties, we can form one equation that will describe the properties of a binomial expansion.
(x+y)^n = x^n + nx^(n-1) * y + (n(n-1)/2!)x^(n-2) * y^2 + ... + nxy^(n-1) + y^n
This equation represents the Binomial Theorem.
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