Polynomials

Polynomials are algebraic expressions that contain indeterminates and constants. You can think of polynomials as a dialect of mathematics. They 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, 2x + 9 and x² + 3x + 11 are polynomials.

The degree of the polynomials is determined by the highest exponent. Various operations like addition, subtraction, multiplication and division can be applied on polynomials.

Polynomials are mathematical expressions made up of variables and constants by using arithmetic operations like addition, subtraction, and multiplication. They represent the relationship between variables.

In polynomials, the exponents of each of the 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, but we cannot perform division operations by a variable in polynomials.

Polynomial Examples

Let us understand this by taking an example: 3x² + 5. In the given polynomial, there are certain terms that we need to understand. Here, x is known as the variable. 3 which is multiplied to x² has a special name. We denote it by the term "coefficient". 5 is known as the constant. The power of the variable x is 2.

Below given are a few expressions that are not examples of a polynomial.

The following image shows all the terms in a polynomial.

The highest or greatest exponent of the variable in a polynomial is known as the degree of a polynomial. The degree is used to determine the maximum number of solutions of a polynomial equation (using Descartes' Rule of Signs).

Example 1: A polynomial 3x⁴ + 7 has a degree equal to four.

The degree of the polynomial with more than one variable is equal to the sum of the exponents of the variables in it.

Example 2: Find the degree of the polynomial 3xy.

In the above polynomial, the power of each variable x and y is 1. To calculate the degree in a polynomial with more than one variable, add the powers of all the variables in a term. So, we will get the degree of the given polynomial (3xy) as 2.

Similarly, we can find the degree of the polynomial 2x²y⁴ + 7x²y by finding the degree of each term. The highest degree would be the degree of the polynomial. In this example:

• The degree of 2x²y⁴ is 2 + 4 = 6 and
• The degree of 7x²y is 2 + 1 = 3
• Among these, 6 is the bigger number and hence it is the degree of the polynomial 2x²y⁴ + 7x²y.

The standard form of a polynomial refers to writing a polynomial in the descending power of the variable.

Example: Express the polynomial 5 + 2x + x² in the standard form.

To express the above polynomial in standard form, we will first check the degree of the polynomial.

• In the given polynomial, the degree is 2. Write the term containing the degree of the polynomial.
• Now, we will check if there is a term with the exponent of variable less than 2, i.e., 1, and note it down next.
• Finally, write the term with the exponent of the variable as 0, which is the constant term.

Therefore, 5 + 2x + x² in standard form can be written as x² + 2x + 5.

Always remember that in the standard form of a polynomial, the terms are written in decreasing order of the power of the variable, here, x.

The terms of polynomials are defined as the parts of the expression that are separated by the operators "+" or "-". For example, the polynomial expression 2x³ - 4x² + 7x - 4 consists of four terms. The terms are classified into two types: like terms and unlike terms.

Like terms in polynomials are those terms which have the same variable and same power. Terms that have different variables and/or different powers are known as unlike terms. Hence, if a polynomial has two variables, then all the same powers of any ONE variable will be known as like terms. Let us understand these two with the help of examples given below.

For example, 2x and 3x are like terms. Whereas, 3y⁴ and 2x³ are unlike terms.

Polynomials can be categorized based on their degree and their power. Based on the numbers of terms, there are mainly three types of polynomials that are listed below:

• Monomials
• Binomials
• Trinomials

Monomial is a type of polynomial with a single term. For example, x, -5xy, and 6y². A binomial is a type of polynomial that has two terms. For example, x + 5, y² + 5, and 3x³ - 7. While a Trinomial is a type of polynomial that has three terms. For example 3x³ + 8x - 5, x + y + z, and 3x + y - 5. However, based on the degree of the polynomial, polynomials can be classified into 4 major types:

• Zero Polynomial: It a polynomial whose degree is equal to zero.
• Constant polynomial: It is a polynomial with just a constant and no variables. For example 3, 5, or 8, etc.
• Linear polynomial: It is a polynomial with degree 1. Example: x + y - 4.
• Quadratic polynomial: It is a polynomial with degree 2. Example: 2p² - 7.
• Cubic polynomial: It is a polynomial with degree 3. Example: 6m³ - mn + n² - 4.

A polynomial expression has terms connected by the addition or subtraction operators. There are different properties and theorems on polynomials based on the type of polynomial and the operation performed. Some of these are as given below,

Theorem 1: If A and B are two given polynomials then,

• deg⁡(A ± B) ≤ max(deg⁡ A, deg ⁡B), with the equality if deg⁡ A ≠ deg ⁡B
• deg⁡(A⋅B) = deg⁡ A + deg⁡ B

Theorem 2: Given polynomials A and B ≠ 0, there are unique polynomials Q (quotient) and R (residue) such that,

A = BQ + R and deg R < deg B

Theorem 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.

Theorem 4: If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.

Theorem 5: Polynomial P(x) of degree n > 0 has a unique representation of the form P(x) = k(x - x₁)(x - x₂)...(x - x_n), where k ≠ 0 and x₁,…,x_n are complex numbers, not necessarily distinct.

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

Theorem 6: Polynomial of n-th degree has exactly n complex/real roots along with their multiplicities.

Theorem 7: If a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by Q⋅R.

Theorem 8: If ß is a complex zero of a real polynomial P(x), then so is \(\overline{ß}\) (complex conjugate of ß).

Theorem 9: A real polynomial P(x) has a unique factorization (up to the order) of the form,

P(x) = (x - r₁)...(x - r_k)(x² - p₁x + q₁)...(x² - p_lx + q_l),

where r_i and p_j, q_j are real numbers with p_i² < 4qi and k + 2l = n.

Theorem 10 (Remainder Theorem): The remainder when a polynomial f(x) is divided by (x - a) is f(a).

Theorem 11 (Rational Root Theorem): A rational root of a polynomial functional f(x) = a_n xⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ is of the form p/q, where p is a factor of a₀ and q is a factor of a_n. This theorem is very helpful in finding the rational zeros of a polynomial.

The basic algebraic operations can be performed on polynomials of different types. These four basic operations on polynomials can be given as,

• Addition of polynomials
• Subtraction of polynomials
• Multiplication of polynomials
• Division of polynomials

Addition of Polynomials

Addition of polynomials is one of the basic operations that we use to increase or decrease the value of polynomials. Whether you wish to add numbers together or you wish to add polynomials, the basic rules remain the same. The only difference is that when you are adding, you align the appropriate place values and carry the operation out. However, when dealing with the addition of polynomials, one needs to pair up like terms and then add them up. Otherwise, all the rules of addition from numbers translate over to polynomials. Have a look at the image given here in order to understand how to add any two polynomials.

Subtraction of Polynomials

As discussed above, the rules for the subtraction of polynomials are very similar to subtracting two numbers. To subtract a polynomial from another, we just add the additive inverse of the polynomial that is being subtracted to the other polynomial. Another easy way to subtract polynomials is, just to change the signs of all the terms of the polynomial to be subtracted and then add the resultant terms to the other polynomial as shown below. We just have to align the given polynomials based on the like terms.

Multiplication of Polynomials

The multiplication operation on polynomials follows the general properties like commutative property, associative property, distributive property, etc. Applying these properties using the rules of exponents we can solve the multiplication of polynomials. To multiply to polynomials, we just multiply every term of one polynomial with every term of the other polynomial and then add all the results. Here is an example to multiply polynomials.

For example, (2x + 3y)(4x - 5y) = 2x(4x - 5y) + 3y(4x - 5y) = 8x² - 10xy + 12xy - 15y²

⇒ (2x + 3y)(4x - 5y) = 8x² + 2xy - 15y²

Division of Polynomials

The division of polynomials is an arithmetic operation where we divide a given polynomial by another polynomial which is generally of a lesser degree in comparison to the degree of the dividend. There are two methods to divide polynomials.

• Long division of polynomials
• Synthetic division

To learn more about each type of division, click on the respective link.

Factorization of polynomials is the process by which we decompose a polynomial expression into the form of the product of its irreducible factors, such that the coefficients of the factors are in the same domain as that of the main polynomial. There are different techniques that can be followed for factoring polynomials, given as,

• Method of Common Factors
• Grouping Method
• Factoring by Splitting Terms
• Factoring Using Algebraic Identities

Based on the complexity of the given polynomial expression, we can follow any of the above-given methods.

Polynomial Equations

A polynomial equation is an equation formed with variables, exponents, and coefficients together with operations and an equal sign. The general form of a polynomial equation is P(x) = a_n xⁿ + . . + a₁x + a₀. Some examples of polynomial equations are x² + 3x + 2 = 0, x³ + x + 1 = 0, x + 7 = 0, etc.

Solving polynomial equations is to find the value of the variable that satisfies the equation.

Polynomial Functions

The general expressions containing variables of varying degrees, coefficients, positive exponents, and constants are known as polynomial functions. In other words, a polynomial function is a function whose definition is a polynomial. Here are some examples of polynomial functions:

• f(x) = x² + 4
• g(x) = -2x³ + x - 7
• h(x) = 5x⁴ + x³ + 2x²

Solving a polynomial means finding the roots or zeros of the polynomials. We can apply different methods to solve a polynomial depending upon the type of the polynomial, whether it is a linear polynomial, quadratic polynomial, and so on. Let us first understand what is meant by the zero of a polynomial.

Zeros of Polynomials

The roots or zeros of polynomial are the real values of the variable for which the value of the polynomial would become equal to zero. So, if we say any two real numbers, ‘α’ and ‘ß’ are zeroes of polynomial p(x), then p(α) = 0 and p(ß) = 0. For example, for a polynomial, p(x) = x² - 2x + 1, we observe, p(1) = (1)² - 2(1) + 1 = 0. Therefore, 1 is a zero or root of the given polynomial. This also means that (x - 1) is a factor of p(x).

Now, to find the zero or root of any polynomial, that is, to solve any polynomial, we can apply different methods,

• Factorization
• Graphical Method
• Hit and Trial Method

Important Notes on Polynomials:

• Terms in a polynomial can be only separated by the '+' or '-' sign.
• For any expression to become a polynomial, the power of the variable should be a whole number.
• The addition and subtraction of a polynomial are possible between like terms only.
• All the numbers in the universe are called constant polynomials.

☛Related Articles:

• Polynomials in One Variable
• Linear Equations
• Polynomial Solver Calculator

What is Polynomial Meaning?

Polynomial is an algebraic expression with terms separated using the operators "+" and "-" in which the exponents of variables are always nonnegative integers. For example, x² + x + 5, y² + 1, and 3x³ - 7x + 2 are some polynomials.

What are Coefficients in a Polynomial?

The coefficients of a polynomial are multiples of a variable or variable with exponents. Let us take the polynomial 3x³ - 2x + 7, the coefficient of x³ is 3, and the coefficient of x is -2.

Is 8 a Polynomial?

8 is a Polynomial. Since the degree of this polynomial is equal to zero, it is an example of a constant polynomial.

What are Monomials, Binomials, and Trinomials?

Monomial is a type of polynomial with a single term. For example x, -5xy, and 6y². While a binomial will be having two terms. For example, x + 5, y² + 5, and 3x³ - 7. While a trinomial is a type of polynomial that has three terms. For example 3x³ + 8x - 5, x + y + z, and 3x + y - 5.

What is Constant in a Polynomial?

A number that is not a multiple of any variable in a polynomial is known as a constant. For example, in the polynomial 4x⁴ + 3x² - 5, -5 is the constant. A polynomial may or may not have a constant in it.

What is Descartes Rule of Signs of Polynomials?

Descartes' rule of signs is used to determine the number of positive/negative real zeros of a polynomial f(x).

• The number of positive real zeros of the f(x) in standard form is the number of sign changes in it and
• the number of negative real zeros of f(x) is the number of sign changes in f(-x).

What is a Polynomial Equation?

A polynomial equation is when two different polynomials are combined together by means of an equal-to sign. In this case, the expression then becomes a polynomial equation.

Why are Polynomials Important?

Polynomials form a large group of algebraic expressions that show the relationship between the variables in them. Any expression with only whole numbers as the powers of the variables is termed polynomials. As they cover such a huge chunk of all algebraic expressions, they tend to have a wide variety of applications.

How to Multiply and Divide Polynomials?

For the multiplication of polynomials, there are three laws that are to be kept in mind - distributive law, associative law, and commutative law. For division, the most common method used to divide one polynomial by another is the long division method.

Is Zero a Polynomial?

Number 0 is a special polynomial called "zero polynomial." Since it is a constant it is also referred to as a constant polynomial.

Where to Find Polynomial Calculator?

We can find the polynomial calculator by clicking here. We can use this to add/subtract/multiply/divide polynomials.