Types of Polynomials: Monomial, Binomial, Trinomial

Polynomials are algebraic expressions that may comprise of exponents which are added, subtracted or multiplied. Polynomials are of different types. Namely, Monomial, Binomial, and Trinomial. A monomial is a polynomial with one term. A binomial is a polynomial with two, unlike terms. A trinomial is an algebraic expression with three, unlike terms. In the following section, we will study about polynomials and types of polynomials in detail.

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What is a Polynomial?

In simple words, polynomials are expressions comprising a sum of terms, where each term holding a variable or variables is elevated to power and further multiplied by a coefficient. Amusingly, the simplest polynomials hold one variable.

 

Types of Polynomials

Let us get familiar with the different types of polynomials. It will form the base to further learning.

1. Monomials – Monomials are the algebraic expressions with one term, hence the name “Mono”mial. In other words, it is an expression that contains any count of like terms. For example, 2x + 5x + 10x is a monomial because when we add the like terms it results in 17x. Furthermore, 4t, 21x²y, 9pq etc are monomials because each of these expressions contains only one term.
2. Binomials – Binomials are the algebraic expressions with two unlike terms, hence the name “Bi”nomial. For example, 3x + 4x² is binomial since it contains two unlike terms, that is, 3x and 4x². Likewise, 10pq + 13p²q is a binomial.
3. Trinomials – Trinomials are the algebraic expressions with three unlike terms, hence the name “Tri”nomial. For example- 3x + 5x² – 6x³ is a trinomial. It is due to the presence of three, unlike terms, namely, 3x, 5x² and 6x³. Likewise, 12pq + 4x² – 10 is a trinomial.

There is another type of polynomial called the zero polynomial. In this type, the value of every coefficient is zero. For example: 0x² + 0x – 0

Degree of a Polynomial

It is simply the greatest of the exponents or powers over the various terms present in the algebraic expression.

Example: Find the degree of  7x – 5

In the given example, the first term is 7x, whereas the second term is -5. Now, let us define the exponent for each term. The exponent for the first term 7x is 1 and for the second term -5, it is 0. Since the highest exponent is 1, the degree of 7x – 5 is also 1.

Polynomial Equation

A single-variable polynomial having degree n has the following equation:

a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x¹ + a₀x⁰

In this, a’s denote the coefficients whereas x denotes the variable. Since x¹ = x and x⁰ = 1 considering all complex numbers x. Therefore, the above expression can be shortened to:

a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a

When an nth-degree of single-variable polynomial equals to 0, then the resultant polynomial equation of degree ‘n’ acquires the following form:

a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a = 0

 

Browse more Topics under Algebraic Expressions And Identities

• Introduction to Algebraic Expressions
• Operations on Algebraic Polynomials
• Standard Identities of Binomials and Trinomials

Solved Examples For You

Question 1: Which of the following is a binomial?

a. 8*a+a           b. 7a² + 8b + 9c

c. 3a*4b* 2c    d. 11a² + 11b²

Answer : d. 11a² + 11b²

a) 8a+a will give 9a which is monomial.
b)7a² + 8b + 9c is a trinomial
c)3a*4b* 2c will give 24abc, which is a monomial
d) 11a²+11b² is a binomial

Question 2: What kind of polynomial has 4 terms?

Answer:  A quadrinomial has 4 terms. However, that merely means it consists of 4 terms.

Question 3: What is a polynomial with 5 terms called?

Answer: Expressions having more than three terms is labelled merely by its number of terms. For instance, a five-term polynomial is a polynomial that has five terms.

Question 4: What is a zero polynomial?

Answer: It is a constant polynomial all of whose coefficients equals 0. The equivalent polynomial function is the constant function that has value 0, which we also call the zero map. Thus, the zero polynomial is said to be the additive identity of the additive group of polynomials.

Question 5: Who invented polynomials?

Answer: Rene Descartes invented polynomials. He is one of those who were responsible for introducing the concept of the graph of polynomial equations in 1637 in La geometric. Moreover, before this, there was multiplication of polynomials happening in the 15^th century where they wrote equations in words.