Polynomials in One Variable

Polynomials in One Variable

In algebra, it is very important to learn about polynomials. 

Look at the basic form of a polynomial.

polynomial

There are many ways in which we can classify polynomials, and one such way is the number of variables present in the polynomial.

Can you tell how many variables are there in the above polynomial?

In this mini-lesson, we will learn about polynomials in one variable including its definition, classification based on degree, and polynomials in two variables through visualization, some solved examples, and a few interactive questions for you to test your understanding.

Lesson Plan

What Is the Definition of Polynomials in One Variable?

In mathematics, a polynomial is an expression that consists of variables and coefficients, involving the operations of addition, subtraction, multiplication, and exponentiation of variables.

The word "polynomial" contains two words, namely, “poly" and “nomials”. "Poly" means many, and "nomials" means terms. Hence an expression containing many terms is called polynomials, having variables and coefficients. 

Now, let's learn about the definition of polynomial in one variable.

Polynomials in one variable are those expressions in which there is only one variable present.

Some examples of polynomials in one variable are given below:

\[\begin{align} x^2+3x-2 \end{align}\]

\[\begin{align} 3y^3+2y^2-y+1 \end{align}\]

\[\begin{align} m^4-5m^2+8m-3 \end{align}\]

 
tips and tricks
Tips and Tricks
  1. Before learning about polynomials in one variable, it is important to learn what do we consider as polynomials.
  • Polynomials should have a whole number as the degree. Expressions with negative exponents are not polynomials. For example, \(x^{-2}\) is not a polynomial.
  • Polynomials do not have a variable in the denominator. For example, \(\dfrac{2}{x+2}\) is not a polynomial.

Classification of Polynomials in One Variable Based on Degree

To learn about the classification of one variable polynomial based on degree, let us first understand the meaning of the degree of a polynomial.

The highest power of the variable in a polynomial is called the degree of the polynomial. For example, in the following equation: \(x^2+2x+4\), the degree of the polynomial is \(2\), i.e., the highest power of the variable in the polynomial.

For a multivariable polynomial, it is the highest sum of powers of different variables in any of the terms in the expression.

Based on the degree of a polynomial in one variable, it can be classified into 4 types:

  • Zero or constant polynomial
  • Linear polynomial
  • Quadratic polynomial
  • Cubic polynomial 

types of polynomial in one variable based on degree

Zero or Constant Polynomial

A polynomial having 0 as the degree of the polynomial is termed as zero or constant polynomial. Such polynomials only have constant terms with no variable.

\(2\), that we can also write as \(2x^0\), is an example of zero polynomial.

Linear Polynomial

A polynomial with 1 as the degree of the polynomial is termed as linear polynomial.

Linear polynomials can have multiple variables. 

So, we consider linear polynomial in one variable as a type of polynomial in one variable.

These polynomials have only one solution.

Some examples are given below:

\[\begin{align} x+7 \end{align}\]

\[\begin{align} 3y-27 \end{align}\]

\[\begin{align} 7z+\dfrac{17}{2} \end{align}\]

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2, i.e, the highest exponent of the variable is 2

Quadratic polynomials in one variable have only two solutions possible.

Some examples are given below,

\[\begin{align} x^2+7x+12 \end{align}\]

\[\begin{align} 2m^2-5m+3 \end{align}\]

\[\begin{align} 8y^2-2 \end{align}\]

Cubic Polynomial

A cubic polynomial is a polynomial of degree 3, i.e, the highest exponent of the variable is 3

Cubic polynomials in one variable have only three solutions possible.

Some examples are given below,

\[\begin{align} 4y^3-8 \end{align}\]

\[\begin{align} x^3+5x^2-6x-4 \end{align}\]

\[\begin{align} 5z^3-\dfrac{3}{4} \end{align}\]


What Do You Mean by Polynomials in Two Variables?

Polynomials in two variables are expressions with two variables present in them. For example, \(x+y=5\). 

In polynomials in two variables, both the variables are dependent on each other.

Some examples of polynomials in two variables are given below.

\[\begin{align} x^3+y^3-xy=7 \end{align}\]

\[\begin{align} mn+m^2-8=0 \end{align}\]

\[\begin{align} 3p^5+q^4-32=0 \end{align}\]

\[\begin{align} a-3b=9 \end{align}\]

Like polynomials in one variable, polynomials in two variable can also be classified as linear polynomials, quadratic polynomials, and cubic polynomials.

 
important notes to remember
Important Notes
  1. A variable with no exponent is considered to be raised to the power 1. For example, in the polynomial \(2x^2+3x\), 3x has the exponent value as 1.
  2. A constant's exponent is always raised to the power 0, which makes it equal to 1. For example, in the polynomial \(4x+ 6\), 6 is considered as 6 x\(^0\) and forms the last term of the polynomial.

Now, look at some polynomial in one variable examples for a deeper understanding.

Solved Examples

Example 1

 

 

Find the perimeter of the given square. Also, state the type of polynomial (based on degree) obtained.

state the type of polynomial

Solution

We know that the perimeter of a square is the sum of all its sides.

In a square, all sides are equal. 

So, if we add all four sides of the square, we get,

\(8x+8x+8x+8x\), which equals to \(32x\).

\(\therefore\) Perimeter of the square is \(32x\) units.

\(32x\) has only one variable with degree 1.

So, it is a linear polynomial in one variable.

\(\therefore\) The perimeter of the square is \(32x\) units which is a linear polynomial in one variable.
Example 2

 

 

Classify the given polynomials based on their degrees.

classify the given polynomials

Solution

There are 6 polynomials given on the board.

Let's classify each one of them into zero, linear, quadratic, and cubic polynomials.

\(n^3+6\) is a cubic polynomial in one variable as the highest exponent (degree of polynomial) with the variable is 3

\(5x^2-2x+1\) is a quadratic polynomial in one variable as the degree is 2

\(p-82\) is a linear polynomial in one variable as the degree of polynomial is 1

\(2p^2+q-11\) is a quadratic polynomial in two variables. The degree of the polynomial is 2 and there are two variables present in it, \(p\) and \(q\).

\(34\) is a zero polynomial as the degree of polynomial is 0


Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of polynomials in one variable. The math journey around polynomials in one variable starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Polynomials in One Variable

1.What is the meaning of polynomials in one variable?

Polynomials in one variable are those expressions in which there is only one variable present.

2.What are the examples of polynomials in one variable?

Some examples of polynomials in one variable are: \(3x^2+4x-7\),  \(5y-7\), \(7z^3-10\).

3.How do you find the polynomial in one variable?

To find whether a polynomial is in one variable, we just have to see how many variables are present in the expression. If there is only one variable, let's say \(x\) in the polynomial, we consider it as a polynomial in one variable.

4.How many terms can a quadratic polynomial in one variable have?

A quadratic polynomial in one variable can have a minimum of one term and a maximum of 3 terms. The standard form of quadratic polynomial in one variable is \(Ax^2+Bx+C\).

5.How many terms can a cubic polynomial in one variable have?

A cubic polynomial in one variable can have a minimum of one term and a maximum of 4 terms. The standard form of cubic polynomial in one variable is \(Ax^3+Bx^2+Cx+D\).

6.What is an example of polynomials in two variables?

An example of a polynomial in two variables is \(x+y-7\).

7.What is the degree of polynomial 7?

The degree of polynomial 7 is 0, as we can rewrite 7 as \(7x^0\).

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