Polynomials| Definition, Properties and Examples

Polynomials are mathematical expressions that are used in various fields of mathematics, astronomy, economics, etc. Based on number terms, there are different types of polynomials such as monomials, binomials, trinomials, etc.

In this article, we will learn about, Polynomials their definition, Examples, Degrees of Polynomials, Zeros of Polynomials, Polynomial Equations and some example problems.

What are Polynomials?

A Polynomial is an algebraic expression that consists of variables (also known as indeterminates), coefficients, and exponents which are combined with operations of addition, subtraction, multiplication, and the power of variables is non-negative integers.

Polynomial [Tex]P(x)=a_{n}x^n+a_{n−1}x^{n−1}+⋯+a_1x+a_0 ​ [/Tex]

• [Tex]a_n​,a_{n−1}​,…,a_1​,a_0[/Tex]​ are coefficients (which can be real or complex numbers).
• x is the variable.
• n is a non-negative integer representing the degree of polynomial.
• [Tex]a_n[/Tex]is the leading coefficient and [Tex]a_0[/Tex] is the constant term.

Each term consists of a variable raised to a non-negative whole number power, multiplied by a number. In the image added below, we have shown a polynomial, with variables, constants, and leading coefficient.

Characteristics of Polynomials

A polynomial can have the following characteristics:

1. Variables and Exponents: A polynomial consists of one or more variables, such as x or y, raised to non-negative integer exponents. 
2. Coefficients: Coefficients are constants in the polynomial that are multiplied by the variable, These constants can be real or complex but generally these are integers or fractions.
3. Terms: A polynomial is composed of multiple terms, each separated by addition or subtraction operators. Each term consists of a coefficient multiplied by a variable raised to a specific exponent.
4. Leading Term: The leading term of a polynomial is the term with the highest degree. It determines the dominant behavior of the polynomial as the input values increase or decrease.
5. Constant Term: The constant term of a polynomial is a term that does not have any variables. It is the term with zero exponents, and its coefficient represents the y-intercept of the polynomial when graphed.

Polynomials Examples

Various examples of the polynomial equations are:

• 3x³ + 5x² – 4x,
• x² + 2x,
• 5x + y
• √2(x) + y, etc.

Check whether 3x² + 2x – 5 is polynomial or not.

• First term is 3x², where the variable “x” is raised to the power of 2 and multiplied by the coefficient 3.
• Second term is 2x, where the variable “x” is raised to the power of 1 (which is often omitted) and multiplied by the coefficient 2.
• Third term is -5, which does not have a variable since it is a constant term.

As all these follow the definition of polynomial. So this is a polynomial.

Identifying Polynomials

Standard Form of a Polynomial

Standard form of a polynomial is the way of representing a polynomial in mathematics. In standard form of polynomial the terms of the polynomial are written in the descending order of their degree. The example of standard form of a polynomial is:

P(x) = a_nxⁿ + a_n-1x^n-1 +a_n-2x^n-2 + . . .  + a₁x + a₀

The polynomial function can be represented by P(x) where x represents the variable.

Example: Express the variable in Standard form: x² – 3x³ + 4x

• Given Polynomial,

= x² – 3x³ + 4x

In standard form polynomial is arranged in decreasing order of their degree

= -3x³ + x² + 4x

Degree of Polynomial

Degree of polynomial is determined by the highest exponent or power of the variable in the expression. It represents the highest degree term in the polynomial.

To find the degree of polynomial there comes two cases:

Degree of Single Variable Polynomial

The degree of a single variable polynomial is determined by the highest exponent or power of that variable in the polynomial expression. It is represented by the highest degree term in the polynomial.

Example: 5x⁵ – 2x³ + 4x² – x + 4

Solution:

The highest exponent is 5, which corresponds to the term 5x⁵. Therefore, the degree of this polynomial is 5.

Note: The degree of any polynomial which contains constant terms will be zero.

Degree of a Multivariable Polynomial

The degree of a multivariable polynomial is determined by the sum of the exponents of the variables in each term, considering the highest sum among all the terms.

Example: 5x²y³ – 2xy⁴ + 3x³+ 10

Solution:

In this expression, we have terms with variables x and y raised to different powers. To find the degree, we calculate the sum of the exponents for each term:

• First term, 5x²y³, has a sum of exponents equal to 2 + 3 = 5.
• Second term, -2xy⁴, has a sum of exponents equal to 1 + 4 = 5.
• Third term, 3x³, has a sum of exponents equal to 3.

Among all the terms the highest sum of exponents is 5. Therefore, the degree of this polynomial is 5.

Terms in a Polynomial

In a polynomial various terms are separated using the plus(+) and the minus(-) sign. Depending on the number of terms in a polynomial a polynomial can be monomial, binomial, etc.

In a polynomial we classifies various terms into like terms and unlike terms.

• Like terms are the terms that have same variable and with same exponent.
• Unlike terms are terms that have different variables or different powers(even with same variable).

Read more about Like and Unlike Terms.

For example, Identify like terms and unlike terms in the polynomial, 4x³ + 2x² -3xy + x³ – xy

Given Polynomial,

• 4x³ + 2x² -3xy + x³ – xy

Like Terms: (4x³, x³) and (-3xy, -xy)

Unlike Terms: (4x³, x²), (x³, x²), (-3xy, x²), (-xy, x²), etc

Types of Polynomials

Polynomials are classified based on the number of terms they contain, and they can be divided into the following categories:

Monomial: A polynomial with only one term is called a Monomial, and it must not be zero.

Examples: 2x, 5x³, 10

Binomial: A polynomial with two terms is called a binomial. It can be expressed as the sum or difference of two or more monomials.

Examples: 3x⁴ + 4x, 5y³ + 3x, xy³ + 3y

Trinomial: A polynomial with three terms is referred to as a trinomial.

Examples: 5x² + 4x + 10, 10y⁴ + 4x⁴ + 2x², 7y² + 3y + 10

Quadronomial: Polynomial containing 4 terms is called the Quadronomial.

Example: 3x² + 2x + y³ – 11y, 13x³ – x + y² – 4y, etc.

Properties of Polynomials (Theorems of Polynomials)

Various properties of the polynomials are:

For two polynomials P(x) and Q(x)

• Degree (P ± Q) ≤ max(deg⁡ P, deg ⁡Q)
• Deg⁡(P⋅Q) = deg⁡ P + deg⁡ Q

For two given polynomials P(x) and Q(x) we always have unique polynomials Q (Quotient Polynomial) and R (Residue Polynomial) such that,

• P = R.Q + R

If a polynomial P(x) is divisible by polynomial x – a, then P(a) = 0 is always true. This is also called Bezout’s Theorem or Factor Theorem.

If a polynomial Q divides the polynomial P then zeros of polynomial Q are the zeros of polynomial P.

For a polynomial of n degree we have n roots real or complex. Remainder of a polynomial f(x) when divided by (x – a) is f(a).

Operations on Polynomials

Addition of Polynomials

When adding polynomials, it is important to combine the like terms, which means adding the terms that have the same variable and exponent. Adding polynomials will always yield a polynomial of the same degree as the original polynomials being added.

Example: Add the polynomials 3x² + 2x + 1 and 2x² – 4x + 3.

Solution:

Identify like terms,

• x² are 3x² and 2x² are like terms.
• x are 2x and -4x are like terms.
• Constant terms are 1 and 3.

Combining the like terms, we have and add the coefficients of the like terms

(3x² + 2x²) + (2x – 4x) + (1 + 3)

Simplifying,

5x² – 2x + 4

Therefore,

(3x² + 2x + 1) + (2x² – 4x + 3)

= 5x² – 2x + 4

Subtraction of Polynomials

When it comes to subtracting polynomials, the process is similar to addition, but with a different operation. You subtract the like terms to find the solution. It’s important to remember that subtracting polynomials will always result in a polynomial of the same degree.

Example: Subtraction the polynomial 2x² + 3x – 5 from the polynomial 4x² – 2x + 7.

Solution:

(4x² – 2x + 7) – (2x² + 3x – 5)

= 4x² – 2x + 7 – 2x² – 3x + 5

= (4x² – 2x²) + (-2x – 3x) + (7 + 5)

= 2x² – 5x + 12

Multiplication of Polynomials

When multiplying polynomials with another polynomial, the resulting polynomial will generally have a higher degree than the original polynomials, unless one of them is a constant polynomial.

Example: Multiply the polynomials (x + 2) and (x – 3).

Solution:

= (x + 2) × (x – 3)

= x × x + x × (-3) + 2 × x + 2 × (-3)

= x² – 3x + 2x – 6

= x² – x – 6

Division of Polynomials

It is an arithmetic operation by which a polynomial is divided by another polynomial in a known as polynomial division. For this operation to succeed, the divisor polynomial must have a degree that is less than or equal to the dividend polynomial.

There are several ways to divide polynomials, some of which include:

• Long Division
• Synthetic Division
• Polynomial Division Using Factors

Read more about Dividing Polynomials

Factorization of Polynomials

Factorization of polynomials refers to the process of breaking down a polynomial expression into a product of simpler polynomial expressions. It involves finding the factors of the given polynomial, which are the polynomial expressions that, when multiplied together, yield the original polynomial.

Let’s first understand what factors are before learning about the techniques for factoring polynomials. Factors are just multiples of numbers that produce the initial needed number. In the same way, when it comes to polynomials, the factors are other polynomials that were multiplied together to produce the original polynomial.

Now let’s understand the numerous methods for factoring polynomials and the most used formulae for doing so.

There are four different factoring polynomials formulas as follows:

• Greatest Common Factor (GCF)
• Substitution Method
• Grouping Method
• Difference of Two Squares Identity

Greatest Common Factor (GCF)

This is the basic method for factoring polynomials as Greatest Common Factor (GCF). In this method, we have to identify any common factor in all the terms. If any common factor is found then factor it out of the polynomial. It is simply a reverse procedure of the distributive law.

In the case of distributive law, we get:

a(b+c) = ab + ac

Whereas in the case of factorization, we invert the process

ab + ac = a(b+c)

Here ‘a’ is the greatest common factor.

Example: Factorize 2x² + 4x

Solution:

= 2x² + 4x

= 2(x² + 2x)

Substitution Method

This method can be very helpful if a given polynomial is too complex, in such case, we have to figure it out and try substituting the complicated terms with a simpler term to solve. Therefore it makes it much easier to factor out.

Example: Factorize 3x² + 12xy + 9y²

Solution:

Let’s substitute A = 3x and B = 3y

3x² + 12xy + 9y²

= (A)² + 2AB + (B)²

= A² + 2AB + B²

= (A + B)²

= (3x + 3y)²

Grouping Method

If an expression has an even number of terms but no common factors exist for any of the terms, we can pair the terms together and get the common factor for each pair:

Example: Factorize 3x + 4ay – 3y – 4ax

Solution:

= 3x – 3y + 4ay – 4ax

= 3(x – y) + 4a(y – x)

= 3(x – y) – 4a(x – y)

= (3 – 4a)( x – y)

Difference of Two Squares Identity

This is a specific technique used when dealing with polynomials that can be expressed as the difference of two perfect squares. The identity states that the expression a² – b² can be factored as (a + b)(a – b).

Example: Factorize: 16x² – 25

Solution:

= 16x² – 25

= (4x)² – (5)²

Comparing with a² – b² = (a – b)(a + b)

= (4x + 5)(4x – 5)

Zeros of Polynomial

Zeros of a polynomial, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. In other words, they are the values of x for which the polynomial evaluates to zero.

If x is a zero of a given polynomial P(x) then p(x) = 0

How to Find Zeros of Polynomials?

The number of zeros that a polynomial contains is determined by the degree of the polynomial equation. Polynomial equations are categorized into various types such as linear equations, quadratic equations, cubic equations, and higher-degree polynomials. Each type of equation is examined separately in order to determine the zeros of the polynomial.

Linear Polynomial

A linear polynomial is a polynomial with degree 1. The standard form is y = ax + b, where a and b are real numbers and a≠0.

Examples of linear polynomials: 2x + 3, a – 6b, y – 12, etc.

Zeroes of a Linear Polynomial

The zero of this equation can be calculated by substituting y = 0, and on simplification, we have ax + b = 0, or x = -b/a.

Quadratic Polynomial

A polynomial of degree 2 is known as a quadratic polynomial. The standard form is ax² + bx + c, where a, b, and c are real numbers and a ≠ 0.

Examples of Quadratic Polynomials: x²+ 3x + 4, 3y²+ 7xy + 4y, etc.

Zeroes of Quadratic polynomial

Quadratic equation of the form x² + x(a + b) + ab = 0 can be factorized as

(x + a)(x + b) = 0,

Where we have x = -a, and x = -b as the zeros of the polynomial. 

And for a quadratic equation of the form ax²+ bx + c = 0, which cannot be factorized, the zeros can be calculated using the formula method, and the formula is

x = [- b ± √(b2 – 2ac) ] / 2a

Cubic Polynomial

A cubic polynomial is a polynomial of degree three. The standard form is ax³+ bx² + cx + d, where a, b, c, and d are real numbers and a≠0.
Examples of a Cubic Polynomials: x³ + 4x² + 7x + 2, 3y³ – 2y²+ 4y – 7, etc.

Zeroes of Cubic Polynomial

The cubic equation of the form y = ax³ + bx² + cx + d, can be factorized by applying the remainder theorem. According to the remainder theorem, if we substitute a smaller value, denoted as αあるふぁ, for the variable x and the resulting value of y is zero (y = 0), then (x – αあるふぁ) is one of the roots of the equation. By dividing the cubic equation by (x – αあるふぁ) using long division, we can obtain a quadratic equation. The quadratic equation can then be solved either through factorization or by using the formula method to find the two desired roots of the equation.

Read More about Solving Cubic Equations.

Higher Degree Polynomial

The higher degree polynomial equation is of the form y = axⁿ+ bx^{n – 1}+cx^{n – 2} + ….. px + q. The remainder theorem can be used to factorize higher-degree polynomials, resulting in a quadratic equation. This quadratic equation can then be further factorized to obtain the two zeroes needed for the polynomial.

Learn More: Factorization of Polynomials

Polynomial Identities

All identities of polynomials which are also called as algebraic identities are provided below:

• (a+b)² = a²+b²+2ab
• (a-b)² = a²+b²-2ab
• (a+b)(a-b) = a²-b²
• (x+a)(x+b) = x²+ x(a+b)+ab
• (a+b)³ = a³+3a²b+3ab²+b³
• (a-b)³ = a³-3a²b+3ab²-b³
• a³+b³ = (a+b)(a²-ab+b²)
• a³-b³ = (a-b)(a²+ab+b²)
• (a+b+c)² = a²+b²+c²+2ab+2bc+2ca

Solving Polynomials

Solving polynomials or solving polynomial equations means finding the zeros or roots which are the values of the variable that make the equation true for a specific polynomial equation. There are various method to find the root of the polynomial equations.

Solving polynomial equations is foundational skill in algebra and it is used in fields ranging from engineering to economics, where relationships defined by polynomials need to be analyzed and used.

Example: Solve x² − 5x + 6 = 0

Solution:

Step 1: Set the Equation to Zero

Ensure your polynomial equation is in the standard form, where one side is zero. x² − 5x + 6 = 0

Step 2: Factor the Polynomial

Factorize the polynomial into simpler binomial factors, if possible.

x² − 5x + 6 = (x−2)(x−3).

Step 3: Apply the Zero Product Property

According to the zero product property, if a product equals zero, then at least one of the factors must be zero.

• (x – 2) = 0 and (x – 3) = 0

Step 4: Solve for the Variable

Solve each equation from the previous step:

• From (x – 2) = 0, x = 2.
• From (x − 3) = 0, x = 3.

Step 5: Check Your Solutions

Substitute your solutions back into the original equation to verify:

Put x=2 in x² − 5x + 6 = 0 ⇒ (2)² − 5(2) + 6 = 4 − 10 + 6 = 0

Put x = 3 in x² − 5x + 6 = 0⇒(3)² − 5(3) + 6 = 9 − 15 + 6 = 0.

Articles related to Polynomials:

• Algebraic Expressions
• Factoring Polynomials
• Quadratic Equations
• Polynomial Formula

Polynomials Class 9 Extra Questions

Question 1: Find the value of x in the polynomial equation 2x² – 5x + 3 = 0.

Solution:

We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a

where a = 2, b = -5, and c = 3.

Substituting the values:

x = (5 ± √(25 – 423)) / 2*2

x = (5 ± √(25 – 24)) / 4

x = (5 ± √1) / 4

So, the solutions are:

x = (5 + 1) / 4 = 6 / 4 = 3/2

and x = (5 – 1) / 4 = 4 / 4 = 1.

Question 2: Factorize the polynomial x² – 4x + 4.

Solution:

We observe that the given polynomial is a perfect square trinomial.

It can be written as (x – 2)².

So, the factored form is (x – 2)(x – 2) or (x – 2)².

Question 3: Find all the roots of the polynomial equation x^3 – 6x^2 + 11x – 6 = 0.

Solution:

We can use synthetic division or polynomial long division to factorize the polynomial and find its roots.

After factorizing, we find that the roots are x = 1, x = 2, and x = 3.

Question 4: Given the polynomial 3x⁴ – 7x³ + 2x² – 5x + 1, find its degree and leading coefficient.

Solution:

The degree of a polynomial is the highest power of the variable present. In this case, the degree is 4. The leading coefficient is the coefficient of the term with the highest power of the variable. Here, the leading coefficient is 3.

Question 5: Simplify the expression (2x² – 3x + 1)(x² + 4x – 2).

Solution:

We use the distributive property to expand the expression:

(2x² – 3x + 1)(x² + 4x – 2) = 2x²(x²+4x−2)−3x(x²+4x−2)+1(x²+4x−2)

= 2x⁴+8x³−4x²−3x³−12x²+6x+x²+4x−2

= 2x⁴ + 5x³ – 15x² + 10x – 2

Polynomials Class 10 Extra Questions

Example 1: Find the degree of 5x⁵ – 2x³ + 4x² – x + 4.

Solution:

Given Polynomial,

• 5x⁵ – 2x³ + 4x² – x + 4

Highest exponent power in the polynomial is 5

Degree of polynomial is 5

Example 2: Add the polynomials 3x² + 2x + 1 and 2x² – 4x + 3.

Solution:

Given polynomials,

• 3x² + 2x + 1
• 2x² – 4x + 3

Adding these two polynomials,

= (3x² + 2x + 1) + (2x² – 4x + 3)

= (3x² + 2x²) + (2x – 4x) + (1 + 3)

= 5x² – 2x + 4

Example 3: Factorize x² + 3x + 2

Solution:

= x² + 3x + 2

= x² + 2x + x + 2

= x(x + 2) +1(x + 2)

= (x + 1)(x + 2)

Example 4: Find the zeros of the polynomial 2x³ + 7x² + 6x.

Solution:

= 2x³ + 7x² + 3x

= x(2x² + 7x + 3)

= x(2x² + 6x + x + 3)

= x{2x(x +3) +1(x +3)}

= x(2x + 1)(x + 3)

Zeros of the given polynomial are,

• x = 0

2x + 1 = 0

• x = -1/2

x + 3 = 0

• x = -3

Check These Resources:

Polynomials Class 9 Notes

Polynomials NCERT Solutions Class 9

Polynomials Class 10 Notes

Polynomials NCERT Solutions Class 10

Practice Problems on Polynomial

Q1. Find the roots of the polynomial equation 2x² – 5x + 7

Q2. Simplify, (x³ + 8)/(x + 2)

Q3. Find the roots of polynomial equation, 2x³ – 2x = 0

Q4. Factorize: x³ – 2x² + 4x – 11 = 0

Conclusion

Polynomials are foundational elements in mathematics, serving as the building blocks for various complex equations and functions. Understanding their structure, types, and properties is essential for solving polynomial equations and finding roots or zeroes of the polynomial.

From basic operations like addition, subtraction, multiplication, and division to more advanced techniques such as factorization and determining zeros, mastering polynomials unlocks numerous applications in fields such as engineering, physics, economics, and beyond. Whether dealing with monomials, binomials, trinomials, or higher-degree polynomials, having a strong grasp of polynomial equations and their solutions is crucial for any aspiring mathematician or scientist.

Polynomial- FAQs

What is a Polynomial?

A polynomial is an expression that involves variables (usually represented by letters) raised to powers and combined using addition, subtraction, and multiplication. The variables can have coefficients (numbers multiplying the variables) attached to them.

What are Examples of Polynomial?

Some examples of polynomials are:

• 3x² + 2x – 1,
• 4y³ – 5y + 2,
• 7a⁴ – 2b², etc.

What is Zero of Polynomial?

Zeros of a polynomial is a the value of the polynomial substituting it in the polynomial results in Zero. Suppose we have a polynomial P(x) then zero of the polynomial is ‘a’ if P(a) is zero.

What is Coefficient of a Polynomial?

In a variable there are various terms and the coefficient of each term is called the coefficient of a polynomial. For example, in a polynomial x² + 2x + 3

• Coefficient of x² is 1
• Coefficient of 2x is 2

What is Leading Coefficient of a Polynomial?

The coefficient of the highest degree term in the polynomial is called the leading coefficient of a polynomial.

What are Monomials, Binomials and Trinomials?

• Monomials are the polynomial with only one term. Example: 3x²
• Binomials are the polynomial with two terms. Example: 3x² + 2x
• Trinomials are the polynomial with three terms. Example: 3x² + 2x + 1

What is Constant in a Polynomial?

The constant term in the polynomial is the term with no variables. Suppose in the example, 3x² + 2x + 1. The constant term is 1.

What are Polynomial Equations?

Equations which contains plynomials are called the polynomial equations various examples of the polynomial equations are,

• 3x³ + 8x – 5 = 0
• x + y + z = 0
• 3x + y – 5 = 0, etc.

The number of polynomials with zeros as ‒2 and 5 is?

There is 1 polynomial with zeros as -2 and 5, which is x²−3x−10.

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