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# Binomial|Definition & Meaning

## Definition

Any polynomial with just two terms is called a binomial, such as 3x + 9, $\mathsf{\sqrt{x}}$ – 3, x^{2}y – 1, etc. The only polynomial shorter than a binomial is a monomial (only 1 term).

Figure 1 – Difference between monomial, binomial, and trinomial

Figure 1 illustrates the difference between **Monomial, Binomial and Trinomial. **

A polynomial called a **binomial has just terms**. As an illustration, x + 2 is a binomial, where x and 2 are two distinct words. Additionally, in this case, x’s exponent, coefficient, and constant are all equal to 1. A binomial is an algebraic statement with two terms that comprises a **variable, a coefficient, exponents, and a constant**.

Using variables and coefficients, polynomials are algebraic expressions. The word **“indeterminates”** is sometimes used to describe variables. For polynomial expressions, **addition, subtraction, multiplication**, and positive integer exponents are all **mathematical operations** that are possible. However, division by variables is not one of them.

Since the Greek words **“polynomial” and “nominal”** both imply “terms,” the phrase “many terms” may be translated from the word polynomial. Any number of terms, but not infinite ones, can be found in a **polynomial.**

Figure 2 – Difference between polynomial and non-polynomial expression

Figure 2 illustrates the difference between Polynomial and non-polynomial expressions.

Binomial is the name for an algebraic expression with only two terms. It is a polynomial with two terms. It is sometimes referred to as the sum or difference of two or more monomials. It is a polynomial’s most basic form.

A **binomial** can be written as a single indeterminate as follows: ax^{m} + bx^{n}. Where m and n are non-negative distinct integers and a and b are the numbers. X can be **a variable or indefinite.**

**Binomials **are stated in the same way in Laurent polynomials; the main distinction is that m and n may have **negative exponents**. Consequently, we may express it as ax^{-m} + bx^{-n}.

**Mathematical Operations on Binomials**

There can be many **mathematical operations** that can be performed on Binomial like Factorization, addition, subtraction, multiplication, raising to the nth power, and conversion to lower-order binomials.

**Factorization of Binomials **

A binomial can **be factored and written** as the sum of the other two. For instance, (x + y)(x – y) may be used to represent x^{2} – y^{2}.

**Addition of Binomials **

Only when two binomials contain **similar words may they be added**. This implies that the variable and **exponent should be the same**. For instance, Let’s think about two equations. 9x^{3} + 6y and 10x^{3} + 4y. Adding the **two solutions** to the equation (10x^{3} + 4y) and (9x^{3} + 6y). The resulting equation is thus = 19x^{3} + 10y.

**Subtraction of Binomials**

When and only when it includes like words, the **addition operation of two binomials** and their subtraction are analogous. Examples include 12x^{3} + 4y and 9x^{3} + 10y. When the **aforementioned polynomials** are subtracted, the result is (12x^{3} + 4y) – (9x^{3} + 10y) = 12x^{3} + 4y – 9x^{3} – 10y. The resulting equation is thus = 3x^{3} – 6y.

**Multiplication of Binomials**

Using the **distributive property**, two binomials are multiplied, resulting in four terms. The FOIL approach is how it’s often known. Since each term of the **first factor** is multiplied by the second factor in this technique, **multiplication is accomplished**. Thus, a trinomial is used to represent the result of multiplying two two-term polynomials.

**Raising to the n’th Power **

When a binomial is raised to the nth power, it may be written as: (x + y)^{n}.

### Conversion to Lower-order Binomials

Any **higher-order binomial** may be factored down to a lower-order one, such as cubes can be factored down to the **products of squares and another monomial.**

**Binomial Theorem**

The term **“binomial theorem” or “binomial expansion”** refers to the algebraic expansion of a binomial’s powers in elementary mathematics. The theory states that the polynomial (x + y)^{n} may be **expanded into a sum comprising terms** of the type ax^{b}y^{c}, where the exponents b and c are non-negative integers with b + c = n and the coefficient an of each term is a particular positive integer dependent **on n and b.**

Figure 3 – Binomial expansion

Figure 3 illustrates the formula for Binomial Expansion.

At least since the **Greek mathematician** Euclid noted the particular case of the binomial theorem with exponent 2 in the fourth century BC, specific instances of the binomial theorem have been known. There is evidence that the **cube binomial theorem** was understood in India by the sixth century AD.

The number of ways you can choose k items from a total set of n without replacing them is expressed as binomial coefficients, which are **combinatorial numbers** of importance to **ancient Indian mathematicians**. The Chandastra by the Indian poet Pingala (about 200 BC), which includes a technique for its resolution, is the first **recorded mention of this combinatorial problem.**

According to our understanding, Al-book, Karaji’s, which was cited by Al-Samaw’al in his “al-Bahir,” contains the earliest statement of **the binomial theorem** as well as the table of binomial coefficients.

Using an early form of **mathematical induction**, Al-Karaji explained the triangular arrangement of the **binomial coefficients** and offered a mathematical proof for **both Pascal’s triangle** and the binomial theorem.

Omar Khayyam was a mathematician and poet from Persia, and it’s likely that he was aware of the formula to higher levels even if much of his mathematical writings had been destroyed. The 13th-century mathematical writings of Yang Hui and Chu Shih-Chieh provide information on **binomial expansions of tiny degrees**.

Although such works are now also widely available, Yang Hui claims that Jia Xian wrote the approach in a far older document from the 11th century.

**Some Examples of ****Binomials**

### Example 1

Solve the following for the value of x:

a) 3x – 9

b) x^{2} – 2x + 1

c) x^{2} – 6x + 9

### Solution

a) In order to evaluate the value of x for the polynomial, we will put this equation equal to zero, i.e.:

3x – 9 = 0

3x = 9

x = 9/3

**x = 3**

b) In order to evaluate the value of x for the polynomial, we will put this equation equal to zero, i.e., then we will factorize:

x^{2} – 2x + 1

x^{2} – x – x + 1

Now we will take x and 1 common from the equation:

x(x – 1) – 1(x – 1)

(x – 1)(x – 1)

**x = 1**

c) In order to evaluate the value of x for the polynomial, we will put this equation equal to zero, i.e., then we will factorize:

x^{2} – 6x + 9

x^{2} – 3x – 3x + 9

Now we will take x and 3 common from the equation:

x(x – 3) – 3(x – 3)

(x – 3)(x – 3)

**x = 3**

### Example 2

Add the following polynomials

3a + 2b + 6c , 2a + 3b and a + 3b + 2c

### Solution

In order to add the polynomials, we will add all the like terms, such as the terms with ‘a’ as a coefficient, the terms with b, and so on:

3a + 2b + 6c + 2a + 3b + a + 3b + 2c

3a + 2a + a + 2b + 3b + 6c + 2c

**6a + 5b + 8c**

*All images were created using GeoGebra.*