Polynomials: Definition, Types, Degrees, Examples

Polynomial is an algebraic expression in which the exponents of all variables are whole numbers. Polynomials also consist of coefficients and variables. It is essential for students to understand the concepts associated with a polynomial to be able to independently solve algebraic equations.

Polynomial is derived from the Greek word. Poly means many and nomial means terms, so together, we can call a polynomial as many terms. In this article, we will learn about polynomial expressions, types of polynomials, the degree of a polynomial, and properties of a polynomial.

Polynomial: Definition

A polynomial is an algebraic expression that has whole numbers as the exponents of the variables.

Example: \(x + 1,\,\,{x^2} – 1\) and \({x^2} + 3x – 5\) are all polynomials.

An algebraic expression consists of variables with exponents, coefficients, and constants that can be combined using basic mathematical operations like addition, subtraction, and multiplication.

Terms Used to Represent Polynomial

The segment of an algebraic expression, which is separated by \( + \) or \(-\) sign, is the term of a polynomial.

Example: The terms of a polynomial \({x^2} + x + 1\) are \({x^2},\,x\) and \(1.\)

Polynomial: Degrees

The degree of a polynomial is defined as the highest power of variable among all terms in a given algebraic expression.

Example: The degree of an expression \({x^3} – 3x\) is \(3.\)

In the above example, the highest power of variable \(x\) among all terms is \(3\). So, we can tell that the degree of the above expression is \(3.\)

Below we have tabulated the degree of polynomials for your reference:

Classification of Polynomials

Based on the number of terms in an expression, we can classify the polynomial as monomial, binomial, trinomial.

1. Monomial: A monomial is a polynomial that consists of only one term.
Example: \(4x,\;3y,\;{x^2},{y^3},3{a^4},\) etc.

2. Binomial: A binomial is a polynomial that consists of two terms.
Example: \(x + 1,\;{x^2} – 1,\;{y^3} + 4,\;a + 3,\;{x^2} + x,\;\) etc.

3. Trinomial: A trinomial is a polynomial that consists of three terms.
Example: \({x^2} + x + 1,\;{x^2} + {y^2} + 2,\;y – 3x + 2,\;\) etc.

Types of Polynomials

Based on the polynomial degree, we can classify the polynomials as constant or zero polynomial, linear polynomial, quadratic polynomial, cubic polynomial, and quartic polynomial.

Constant or Zero Polynomial: A polynomial whose power of the variable is zero is known as a constant or zero polynomial. When the power of the variable is zero, its value is nothing but \(1\) as \({x^0} = 1.\) The zero polynomials will have terms that are constants like \(2,\;5,\;10,\;101,\;\) etc.
Example: \(3{x^0} = 3 \times 1 = 3\)

Linear Polynomial: A polynomial whose power of the variable or the polynomial degree is \(1\) is known as a linear polynomial.
Example: \(x – 1,\;\;y + 1,\;\;a + 4,\) etc.

Quadratic Polynomial: A polynomial whose power of the variable or the polynomial degree is \(2\) is known as a quadratic polynomial.
Example: \({x^2} + x,\;{y^2} + 1,\;\;{a^2} + 8\) etc.

Cubic Polynomial: A polynomial whose power of the variable or the polynomial degree is \(3\) is known as a cubic polynomial.
Example: \({y^3} + 8,\;\;{x^3} – 27,\;\;5 + {a^3},\,{x^3} + {x^2} – x + 2,\;\) etc.

Quartic Polynomial: A polynomial whose power of the variable or the polynomial degree is \(4\) is known as a quartic polynomial.
Example: \({x^4} + {x^3} – {x^2} + x + 1,\,{y^4} – {y^2} + 1\), etc.

Like Terms and Unlike Terms

The terms of a polynomial, in which the variable has the same powers, are known as like terms.

Example: \(2x,\;3x,\;5x,\;x\) and \(3{y^2},\; – 5{y^2},\;{y^2}\) are all like terms.

The terms of a polynomial in which the variables are different and also the same variables with different powers are known as, unlike terms.

Unlike terms are two or more terms that are not like terms. For example, they do not have the same variables or powers.

Example: \(3{x^4}\) and \(2{x^3}\) are unlike terms. Also, \({x^3},\;{y^3}\) are unlike terms.

Addition of Polynomials

The addition is the basic operation that we use to increase the value of a polynomial. The basic rules remain the same as adding the numbers. While adding numbers, we will align the numbers according to their place values and begin the addition operation.
But while adding expressions, we will group the like terms first and carry out an addition of given polynomials.
Example: Add \(3x + 4y\) and \(4x – 5y\)
Solution: Given: \(3x + 4y\) and \(4x – 5y\)
\(\left( {3x + 4y} \right) + \left( {4x – 5y} \right)\)
\( = 3x + 4y + 4x – 5y\)
Grouping the like terms, we get,
\(3x + 4x + 4y – 5y\)
\( = \left( {3 + 4} \right)x + \left( {4 – 5} \right)y\)
\( = 7x – y\)
Therefore, \(3x + 4x + 4y – 5y = 7x – y\)

Subtraction of Polynomials

Subtraction is the basic operation that we use to decrease the value of a polynomial. The basic rules remain the same as subtracting the numbers. However, while subtracting numbers, we will align the numbers according to their place values and find the difference.
But while subtracting the expressions, we will group the like terms first and subtract given polynomials.
Example: Subtract \(2{a^2} – {b^2}\;\) from \({a^2} + {b^2}.\)
Solution: Given: \({a^2} + {b^2} – \left( {2{a^2} – {b^2}} \right)\)
\( = {a^2} + {b^2} – 2{a^2} + {b^2}\)
Grouping like terms, we get
\( = {a^2} – 2{a^2} + {b^2} + {b^2}\)
\(= – {a^2} + 2{b^2}\)
Therefore, \({a^2} + {b^2} – \left( {2{a^2} – {b^2}} \right) = 2{b^2} – {a^2}\)

Solving a Polynomial

While solving a polynomial first and the foremost thing is that we equate the given polynomial expression to \(0.\) After equating, we use different methods to solve the given expression, like solving a polynomial by taking the common factor, factorisation by splitting the middle term, and solving the given polynomials using a standard algebraic formula. So let us see some of the examples of solving linear and quadratic polynomials.

Example 1: Solve \(4x – 16\)
Answer: Equate the given polynomial to \(0.\)
\( \Rightarrow 4x – 16 = 0\)
Taking \(4\) as a common factor, we get
\(\Rightarrow 4\left( {x – 4} \right) = 0\)
\( \Rightarrow x – 4 = 0\)
\( \Rightarrow x = 4\)

Example 2: Solve: \({x^2} + 7x + 12\)
Answer: Given: \({x^2} + 7x + 12\)
We know that the multiples of \(12\) are \(4\) and \(3.\)
So, \({x^2} + 7x + 12 = 0\)
\(\Rightarrow {x^2} + 4x + 3x + 12 = 0\)
\(\Rightarrow \left( {{x^2} + 4x} \right) + \left( {3x + 12} \right) = 0\)
\(\Rightarrow x\left( {x + 4} \right) + 3\left( {x + 4} \right) = 0\)
\(\Rightarrow \left( {x + 4} \right)\left( {x + 3} \right) = 0\)
\(\Rightarrow x + 4 = 0\) and \(x + 3 = 0\)
\(\Rightarrow x = – 4\) and \(x = – 3\)

Factors of a Polynomial

In a similar way of solving the polynomial, we can find the factors of a polynomial. We also use different methods to solve the given expression, like solving a polynomial by taking the common factor, factorization by splitting the middle term and solving the given polynomials using a standard algebraic formula. So let us see some of the examples of solving polynomials.

Example 1: Find the factors of \(3{x^2} + 6x + 4x + 8\)
Answer: Given expression: \(3{x^2} + 6x + 4x + 8\)
Taking \(3x\) as a common factor from the first two terms and \(4\) from the next two, we get.
\(3{x^2} + 6x + 4x + 8 = 3x\left( {x + 2} \right) + 4\left( {x + 2} \right)\)
\(\Rightarrow \left( {x + 2} \right)\left( {3x + 4} \right)\)
Therefore, factors of \(3{x^2} + 6x + 4x + 8\) are \(\left( {x + 2} \right)\left( {3x + 4} \right).\)

Example 2: Find the factors of \({x^2} – 14x + 48\)
Answer: Given expression: \({x^2} – 14x + 48\)
Factorizing the given expression by splitting the middle term, we get
\({x^2} – 6x – 8x + 48\)
\(\Rightarrow x\left( {x – 6} \right) – 8\left( {x – 6} \right)\)
\(\Rightarrow \left( {x – 6} \right)\left( {x – 8} \right)\)
Therefore, factors of \({x^2} – 14x + 48\) are \(\left( {x – 6} \right)\left( {x – 8} \right)\)

Formation of a Polynomial Using Monomial, Binomial and Trinomial

We can form a polynomial by finding the product of different polynomials. Usually, we call it as multiplication of polynomials.
Multiplication is the same as multiplying the numbers. But when multiplying two polynomials, multiply each term in one polynomial with the corresponding term in another polynomial.

Example: \(\left( {2x} \right)\left( {3{x^2}} \right) = 6{x^3},\,\,\left( {2x + 1} \right)\left( x \right) = 2{x^2} + x\)

Polynomials can be formed using the monomials, binomials, and trinomials by the:

1. Multiplication of monomial with a monomial
2. Multiplication of monomial with a binomial
3. Multiplication of monomial with a trinomial
4. Multiplication of binomial with a binomial
5. Multiplication of binomial with a trinomial
6. Multiplication of trinomial with a trinomial and so on

Solved Examples on Polynomials

Q.1. Give one example each of a binomial of degree \(35\) and a monomial of degree \(100.\)
Ans: An example for binomial of degree \(35\) is \({x^{\left( {35} \right)}} + 4x.\)
An example for monomial of degree \(100\) is \({y^{\left( {100} \right)}}.\)

Q.2. Write the degree of each of the following polynomials.
a. \(5{x^3} + 4{x^2} + 7x\)
b. \(3\)
Ans: Degree is the highest power of the variable in the given polynomial. So, the degree of \(5{x^3} + 4{x^2} + 7x\) is \(3,\) and the degree of \(3\) is \(0,\) respectively.

Q.3. Classify the following as linear, quadratic, and cubic polynomial.
a. \({x^2} + x\)
b. \(y – {y^3} + 4\)
c. \(1 + x\)
d. \(3t\)
e. \(x – {x^3}\)
f. \({r^2}\)
g. \(7{x^3}\)
Ans:

Q.4. Add \({x^2} – x + 5\) and \(6{x^2} + 2x – 10.\)
Ans: Given: \(\left( {\;{x^2} – x + 5\;} \right)\& \;\left( {6{x^2} + 2x – 10} \right)\)
Adding them, we get,
\(\Rightarrow {x^2} – x + 5 + 6{x^2} + 2x – 10\)
Grouping the like terms, we get
\(\left( {{x^2} + 6{x^2}} \right) + \left( { – x + 2x} \right) + \left( {5 – 10} \right)\)
\(\Rightarrow 7{x^2} + x – 5\)
So, \(\left( {{x^2} – x + 5} \right) + \left( {6{x^2} + 2x – 10} \right) = \;7{x^2} + x – 5\)

Q.5. Subtract \(2{x^2} – 6x + 12\) from \(3{x^2} – 8x + 7.\)
Ans: Given \(3{x^2} – 8x + 7 – \left( {2{x^2} – 6x + 12} \right)\)
\(\Rightarrow 3{x^2} – 8x + 7 – 2{x^2} + 6x – 12\)
Grouping the like terms, we get
\(\left( {3{x^2} – 2{x^2}} \right) + \left( { – 8x + 6x} \right) + \left( {7 – 12} \right)\)
\(\Rightarrow {x^2} – 2x – 5\)
So, \(\left( {3{x^2} – 8x + 7} \right) – \left( {2{x^2} – 6x + 12} \right) = {x^2} – 2x – 5\)

Summary

Polynomials can be defined as an algebraic expression that has whole numbers as the exponents of the variables. The Polynomials are classified into 5 types namely, Constant or Zero polynomial, Linear polynomial, Quadratic polynomial, Cubic polynomial, and Quartic polynomial. Furthermore, we also learned that the degree of a polynomial is defined as the highest power of variable among all terms in a given algebraic expression.

In this article, we also learned that the first step to solve a polynomial is to equate the given polynomial expression to \(0.\) After equating, we can use several methods to solve the expression. It includes solving the polynomial by the common factor method, factorising by splitting the middle term, etc.

FAQs on Polynomials

Q.1. List out 5 examples of a polynomial?
Ans: The five examples of polynomials are:
\(4x,\;3y,\;{x^2} – 1,\;{y^3} + 4,\;{x^2} + x + 1\)

Q.2. What is the polynomial equation?
Ans:  The equations formed with variables, constants, coefficients, and exponents are called polynomial equations.

Q.3. What are the \(5\) types of polynomials?
Ans: The five types of polynomials are:
a. Constant or Zero polynomial
b. Linear polynomial
c. Quadratic polynomial
d. Cubic polynomial
e. Quartic polynomial

Q.4. How do you identify a polynomial?
Ans: The polynomials are formed with variables, constants, coefficients, and exponents. The degree of a polynomial will always be a whole number. So, below are some of the examples of polynomials.
Example: \(3x,\;y – {y^3} + 4\)

Q.5. Can \(0\) be a polynomial?
Ans: Yes, like any other constant \(0\) can be considered a polynomial and called a zero polynomial.

Q.6. What is Cubic Polynomial?
Ans: The cubic polynomial is defined as a polynomial whose power of the variable or the polynomial degree is \(3\).

Q.7. Define Polynomial Degrees?
Ans: The degree of a polynomial is defined as the highest power of variable among all terms in a given algebraic expression.

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