An algebraic expression that contains only two terms is called a binomial expression. Binomial is a polynomial consisting of only two terms. A binomial can be expressed in the Mathematical form as axm + bxn, where ‘a’ and ‘b’ are the numbers, ‘x’ is an indeterminate or a variable, and ‘m’ and ‘n’ is non-negative and distinctive integers.

If we express the binomials as in Laurent polynomials, then they can be expressed in the form of ax-m + bx-n. The only difference here is that the powers or the exponents can take negative values as well.

Some examples of the binomials are 7a3 + 2y, 4x + 9z, mn4 – mn, etc.

The basic operations that can be performed on binomials are: -

Factorization: Binomials can be factorized as a product of two terms. We can factorize using either the properties or by taking the common terms out. For instance, a2 – b2 can be expressed as (a+b).(a-b).

Addition: Addition of the terms in a binomial can take place only if the terms in the binomial are alike. For example, 2a2 + 5b be the first binomial and 5a2 + 3b be the second binomial, then the sum of these two binomials will give the result as 7a2 + 8b.

Multiplication: When multiplying the binomials, the distributive property is used which multiplies each of the terms of the first binomial with each of the terms of the second binomial using the FOIL method. For example, (ax + b).(cx + d) will be expressed as acx2 + (ad + bc)x + bd.

Subtraction: The subtraction of the binomials take place similar to that of addition. The condition for performing the subtraction of two binomials requires the presence of similar terms. For example, let the first binomial be 6a + 2b and the second binomial be 2a + 3b; therefore, the difference of the two binomials will result in 4a- b.

Raising to The Power n: The binomial raised to the power n can be expressed as (a + b)n, where ‘a’ and ‘b’ are two numbers, and ‘n’ is any positive integer. The value of the binomial raised to the power n is usually calculated using the binomial theorem.

Converting Into Lower-Order Binomials: The binomials of higher order can be converted into lower-order binomials using factorization properties. For example, a3 – b3 can be converted to lower order as (a-b).(a2 + ab + b2).

The binomial term raised to the power n like (a + b)n can be expanded in the form of xamby, where m and y are either positive or zero distinctive integers and m + y= n, and x is a coefficient of the terms of the binomial.

The general formula used in the expansion of a binomial raised to the power n, (x + a)n is given as

\[(x + a)^{n}\] = \[\sum_{k=0}^{n}\] \[\binom{n}{k}\] \[x^{k}\] \[a^{n-k}\]

A few other methods used in the binomial expansion are Pascal’s Triangle, Factorization, Combinations, Binomial theorems and binomial series.

Question 1. Expand (5x – 4)10 using binomial theorem.

Answer: Using binomial theorem,

\[(5x + a)^{10}\] = \[\sum_{k=0}^{10}\] \[\binom{10}{k}\] \[x^{k}\] \[a^{n-k}\]

= (5x – 4)10 = 10C0 (5x)10–0(–4)0 + 10C1 (5x)10–1(–4)1

+ 10C2 (5x)10–2(–4)2 + 10C3 (5x)10–3(–4)3

+ 10C4 (5x)10–4(–4)4 + 10C5 (5x)10–5(–4)5

+ 10C6 (5x)10–6(–4)6 + 10C7 (5x)10–7(–4)7

+ 10C8 (5x)10–8(–4)8 + 10C9 (5x)10–9(–4)9

+ 10C10 (5x)10–10(–4)10

Question 2. Are 2a and 2a + 4 both binomials?

Answer: Binomials are the polynomials having two terms, but 2a only has a single term whereas 2a + 4 has two terms. Therefore, 2a is not a binomial, it is a monomial, and 2a + 4 is a binomial with two terms.

Question 3. What will be the third term in the expansion of (3 + z)5?

Answer: Using the binomial theorem for expansion,

To find the third term, r=3, n=5

So, the third term will be 35-3+1 . z3-1 . {(5*4)/2} = 3z2. 10 = 30z2.

FAQ (Frequently Asked Questions)

1. Why is it important to study the Binomial theorem?

The binomial theorem helps in the expansion of the binomial terms raised to the power n using a simple formula. If binomial theorem were not there, then it would be a very tedious and time-consuming task to calculate the binomial terms raised to the power of more than 10. It would take a person several days to calculate the binomials with higher powers; therefore the binomials theorem holds the significance in mathematics as it helps in reducing the work of calculating higher degrees in just minutes. The binomial theorem is often used by economists to count the probabilities that depend on the various variables and to predict how the economy will vary according to the various concepts in the coming years.

2. How are the binomial theorem and Pascal’s triangle related to each other?

In Pascal’s triangle, each row is built in such a way that it helps in giving us the coefficient of the terms of the binomial in the process of the binomial expansion. To expand a binomial using the Pascal triangle method, you will have to make the entire triangle pattern to find the coefficient of the terms of expansion which can turn out to be really tedious for the binomials with larger exponents. Also, you have to be really focussed while making Pascal’s triangle as the value of each place in the triangle comes from the previous row, and that increases the chance of making mistakes. Whereas, the binomial theorem used the concept of Pascal’s triangle to make the formula using combination and summation that does the work of expanding a binomial really easy and much faster than Pascal’s triangle.