What are Polynomials? (Definition, Types & Examples)

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You know that algebraic expressions are mathematical statements consisting of variables, constants, and operators. You can perform the four mathematical operations – addition, subtraction, multiplication, and division on these algebraic expressions. 

Polynomials are special types of algebraic expressions which are used to express numbers in almost every field of mathematics and are considered very important in certain branches of math, such as calculus. For example, $3x – 5$, $2x^{2} – 3x + 7$, $\frac{2}{9}u^{2} – 4$ are polynomials. Let’s understand what are polynomials, and their different types, and properties.

What are Polynomials in One Variable?

A polynomial in one variable is a type of algebraic expression consisting of terms of the form $a x^{n}$, where $a$ is a real number, $x$ is a variable and $n$ is a positive integer. In other words, a polynomial is an algebraic expression in which the exponents of all variables should be a whole number. The exponents of the variables in any polynomial have to be a non-negative integer. A polynomial comprises constants and variables.

For example, $2x + 9$ is a polynomial in one variable $x$, where $2$ is a coefficient and $9$ is a constant both of which are real numbers. The exponent of $x$ is $1$, which is a positive integer.

Similarly, in the case of $5x^{3} + 7x^{2} – 2x + 8$, the variable is $x$, the coefficients $5$, $7$, $-2$ are all real numbers, the constant term $8$ is also a real number and the powers of the variable $x$, i.e., $3$, $2$ and $1$ are all positive integers.

Standard Form of a Polynomial in One Variable

The standard form of a polynomial refers to writing a polynomial in the descending power of the variable. The general way or standard form of a polynomial is $a_{n}x^{n} + a_{n – 1}x^{n – 1} + a_{n – 2}x^{n – 2} + a_{n – 3}x^{n – 3} + … + a_{1}x + a_{0}$, where

• $a_{n}$, $a_{n – 1}$, $a_{n – 2}$, $a_{n – 3}$, …, $a_{1}$ are called coefficients and are real numbers
• $a_{0}$ is a constant and is a real number
• $n$ is a positive integer

Note: The power of a variable of any term cannot be a negative number or a fractional or a decimal number.

Notation of a Polynomial

A polynomial is represented by a letter followed by the variable in parentheses. 

For example, $p \left(x \right) = x^{2} – 5x + 9$, $q \left(x \right) = 7x – 2$, $r \left(t \right) = \frac{3}{2} t^{3} – 5t + 7$, etc.

Examples

Ex 1: Which of the following expressions are polynomials?

• $2x^{2} – 5x + 9$
• $t^{2} + \sqrt{3}$
• $3\sqrt{y} + y \sqrt{2}$
• $5x – \frac{5}{y}$
• $3s^{2} – 0.5s + 12$
• $x^{0.5} – 2x + 7$

$2x^{2} – 5x + 9$ is a polynomial in one variable $x$, where the coefficients $2$ and $-5$ are real numbers, constant $9$ is a real number and the powers of the variable $2$ and $1$ are positive integers.

$t^{2} + \sqrt{3}$ is a polynomial in one variable $t$, where the coefficient $1$ is a real number, constant $\sqrt{3}$ is a real number and the power of the variable $2$ is a positive integer.

$3\sqrt{y} + y \sqrt{2}$ is not a polynomial. Although the coefficient $3$ is a real number but the power of the variable $y$ is $\frac{1}{2}$ which is not a positive integer.

Note: $\sqrt{y} = y^{\frac{1}{2}} = y^{0.5}$

$5x – \frac{5}{x}$ is not a polynomial. Although the coefficient $5$ is a real number but the power of the variable $x$ is $-1$ which is not a positive integer.

Note: $\frac{5}{x} = 5 x^{-1}$ 

$3s^{2} – 0.5s + 12$ is a polynomial in one variable $s$, where the coefficients $3$ and $-0.5$ are real numbers, constant $12$ is a real number and the power of the variable $2$ is a positive integer.

$x^{0.5} – 2x + 7$ is not a polynomial. Although the coefficients $1$ and $-2$ are real numbers and constant $7$ is also a real number but the power of the variable $x$ is $0.5$ which is not a positive integer.

Terms of a Polynomial

The smaller parts of a polynomial separated by the mathematical operators “$+$” or “$-$” are called the terms of a polynomial. For example, the polynomial expression $9x^{3} – 3x^{2} + 2x – 7$ consists of four terms. The four terms of the polynomial are $9x^{3}$,  $- 3x^{2}$,  $2x$,  and $- 7$.

Like Terms and Unlike Terms

The terms having the same variable and the same power are known as like terms and the terms that have different variables and/or different powers are called the unlike terms. Hence, if a polynomial has two variables, then all the same powers of any one variable will be known as like terms. 

For example, $2x$ and $7x$ are like terms. Whereas, $3y^{4}$ and $4x^{5}$ are unlike terms. 

Note: 

• The two terms can be added or subtracted only when the two terms are like terms.
• The two terms can be multiplied or divided in both the cases, i.e., like or unlike terms

Degree of a Polynomial

The highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. 

For example 

• Degree of a polynomial $p \left(x \right) = 2x – 5$ is $1$
• Degree of a polynomial $q \left(x \right) = \frac{3}{5}x^{2} – 2x + 7$ is $2$
• Degree of a polynomial $p \left(x \right) = 2x – 5$ is $1$
• Degree of a polynomial $p \left(x \right) = 2x – 5$ is $1$

Examples

Ex 1: Find the degree of the polynomial $p \left(m \right) = 3m^{4} – 5m^{2} + 6m + 4$.

The highest power of the variable in the polynomial $p \left(m \right)$ is $4$, therefore, the degree of the polynomial $p \left(m \right)$ is $4$.

Ex 2: Find the degree of the polynomial $p \left(x \right) = \frac {3}{7} x + 9$.

The highest power of the variable in the polynomial $p \left(x \right)$ is $1$, therefore, the degree of the polynomial $p \left(x \right)$ is $1$.

Ex 3: Find the degree of the polynomial $p \left(x \right) = 12$.

The polynomial  $p \left(x \right)$ has only term and it’s a constant with no variable. Therefore, therefore, the degree of the polynomial $p \left(x \right)$ is $0$.

Types of Polynomials

The polynomials are categorized in either of the following ways.

• Based on the number of terms of a polynomial
• Based on the degree of a polynomial

Types of Polynomials Based on the Number of Terms

Based on the number of terms a polynomial has, it can be

• Monomial: Monomials are polynomials having only one term. For example $2x$, $3y$, $5x{^3}$ are all monomials.
• Binomial: Binomials are polynomials having two terms. For example $5x^{2} + 7$, $3x^{3} + 9$, $5x{^5} – 11$ are all binomials.
• Trinomial: Trinomials are polynomials having three terms. For example $x^{2} – 2x + 1$, $5x^{3} + 7x^{2} + 9$ are all trinomials.

Based on the degree of the polynomial, polynomials can be classified into five major types:

• Zero Polynomial: A polynomial is called a zero polynomial if it has only one term and that is zero ($0$). 
• Constant Polynomial: A polynomial having only one term consisting of a constant or a number is called a constant polynomial. All numbers are constant polynomials. Examples of constant polynomial are $5$, $-9$, $\frac {7}{9}$, etc.
• Linear Polynomial: A polynomial whose degree is $1$ is called a linear polynomial. When such a polynomial is graphed we get a straight line. That’s the reason we call such a polynomial a linear polynomial. These polynomials are also known as degree-$1$ polynomials.
• Quadratic Polynomial: A polynomial whose degree is $2$ is called a quadratic polynomial. When such a polynomial is graphed we get a parabola. These polynomials are also known as degree-$2$ polynomials. 
• Cubic Polynomial: A polynomial whose degree is $3$ is called a cubic polynomial. These polynomials are also known as degree-$3$ polynomials.

Examples

Ex 1: Classify the given polynomials as monomial, binomial, or trinomial. Also, find which of these are constant and zero polynomials.

 $2x^{3} + 3x^{2} + 2$, $6x + 7$, $92$, $2x^2 + 5x + 3$, $7x$, $-\frac{1}{2} x^{2} + 11$, $0$, $\frac{x}{2}$

$2x^{3} + 3x^{2} + 2$ has $3$ terms, so it’s a trinomial.

$6x + 7$ has $2$ terms, so it’s a binomial.

$92$ has only one term, so it’s a monomial.

$2x^2 + 5x + 3$ has $3$ terms, so it’s a trinomial. 

$7x$ has only $1$ term, so it’s a monomial.

$-\frac{1}{2} x^{2} + 11$ has only $1$ term, so it’s a monomial. 

$0$ has only $1$ term and that is $0$, so it’s a zero polynomial.

$\frac{x}{2}$ has only $1$ term, so it’s a monomial.

Properties of Polynomials

A polynomial is an algebraic expression connected by mathematical operators “$+$”, or “$-$”. There are different properties of polynomials based on the type of polynomial and the operation performed. Some of the important properties of polynomials are

Property 1: If $p(x)$ and $q(x)$ are two given polynomials then,

• $ \text{deg} ⁡(p(x) \pm q(x) ) \le \text{max}(\text{deg}⁡ (p(x)), \text{deg}(⁡q(x)))$, with the equality if $ \text{deg }⁡ p(x) \ne \text{deg } q(x)⁡$
• $\text{deg }⁡(p(x)⋅q(x)) = \text{deg⁡ }p(x) + \text{deg⁡ }q(x)$

Property 2: Given polynomials $a(x)$ and $b(x) \ne 0$, there are unique polynomials $q(x)$ (quotient) and $r(x)$ (residue) such that,

$a(x) = b(x)q(x) + r(x)$ and $\text{deg } r(x) \lt \text{deg } b(x)$

Property 3 (Bezout’s Theorem): Polynomial $p(x)$ is divisible by binomial $x − a$, if and only if $p(a) = 0$. This is also known as the factor theorem.

Property 4: If a polynomial $p(x)$ is divisible by a polynomial $q(x)$, then every zero of $q(x)$ is also a zero of $p(x)$.

Property 5: Polynomial $p(x)$ of degree $n \gt 0$ has a unique representation of the form $p(x) = k(x – x_1)(x – x_2)…(x – x_n)$, where $k \ne 0$ and $x_1$,…,$x_n$ are complex numbers, not necessarily distinct.

Therefore, $p(x)$ has at most deg $⁡p(x) = n$ different zeros.

Property 6: Polynomial of $n^{th}$ degree has exactly n complex/real roots along with their multiplicities.

Property 7: If a polynomial $p(x)$ is divisible by two coprime polynomials $q(x)$ and $r(x)$, then it is divisible by $q(x)⋅r(x)$.

Property 8: If $\beta$ is a complex zero of a real polynomial $p(x)$, then so is $\overline{\beta}$ (complex conjugate of $\beta$).

Property 9: A real polynomial $p(x)$ has a unique factorization (up to the order) of the form, $p(x) = (x – r_1)…(x – r_k)(x_2 – p_1x + q_1)…(x_2 – p_1x + q_1)$, where $r_i$ and $p_j$, $q_j$ are real numbers with $p_i^2 \lt 4q_i$ and $k + 2l = n$.

Property 10 (Remainder Theorem): The remainder when a polynomial $p(x)$ is divided by $(x – a)$ is $p(a)$.

Practice Problems

1. Which of the these are polynomials: $2x – 6$, $\frac{7x^2 + 3x – 2}{5}$, $2x^{0.5} – 7x$, $x + \frac{1}{x}$, $x^3 – 2x^2 + 5x + 11$, $\sqrt{x} + 4$
2. Find the degree of given polynomials: $2x^2 + 9x + 7$, $3x^6$, $6x – 9$, $5$, $0$
3. State True or False
   • You can add two like terms.
   • You can add two unlike terms.
   • You can subtract two like terms.
   • You can subtract two unlike terms.
   • You can multiply two like terms.
   • You can multiply two unlike terms.
   • You can divide two like terms.
   • You can divide two unlike terms.

FAQs

Conclusion

An algebraic expression of the form $a_{n}x^{n} + a_{n – 1}x^{n – 1} + a_{n – 2}x^{n – 2} + a_{n – 3}x^{n – 3} + … + a_{1}x + a_{0}$, where $a_{n}$, $a_{n – 1}$, $a_{n – 2}$, …, $a_{0}$ are real numbers and $n$ is a positive integer is called a polynomial. The highest power of a variable is called the degree of a polynomial. Based on degree, a polynomial can be linear, quadratic, or cubic, and based on the number of terms in a polynomial, it can be a monomial, binomial, or trinomial.

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