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Table Of Contents

What is Linear Polynomial?

What are the Roots of a Linear Polynomial?

What is the Proof of Roots of Linear Polynomial?

Linear Polynomial Functions

How to Solve Linear Polynomial Function?

What is Zero of Linear Polynomial?

Real Life Applications of Linear Polynomial

Common Mistakes and How To Avoid Them in Linear Polynomial

Solved Examples of Linear Polynomials

FAQs on Linear Polynomial

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Last updated on September 26, 2025

Linear Polynomial

Q: What is a linear polynomial?

A polynomial where the highest power of the variable is 1, then it is called a linear polynomial. 2x + 2 is a linear polynomial.

Q: How many terms can a linear polynomial have?

A linear polynomial can have one or two terms. Example: 3x, 4x + 3 are linear polynomials.

Q: How many roots does a linear polynomial have?

There is exactly one root in a linear polynomial.

Q: What happens if a in the linear polynomial becomes 0?

If a = 0, then it is not a linear polynomial. It becomes a constant.

Q: What is the difference between a linear and a quadratic polynomial?

A quadratic polynomial has the highest degree of 2, and a linear polynomial has the degree of 1.

Professor Greenline Explaining Math Concepts

The algebraic expressions where the variables are raised to a power of non-negative integers are called polynomials. A polynomial where the highest degree of the variable is 1 is known as a linear polynomial. Linear polynomials are the simplest form of polynomials. We will learn more about linear polynomials in this article.

Linear Polynomial for US Students

What is Linear Polynomial?

A linear polynomial is an expression where the variable has the highest power of one. The linear polynomial is of the form p(x) = ax + b, where a and b are real numbers and a is not equal to 0. If a becomes 0, the term with the variable is eliminated, and the equation reduces to just a constant. Some examples of linear equations are: 4x + 8, 8y - 9, etc.

Polynomials are classified into three types based on their degree, they are:

Linear Polynomial

Quadratic Polynomial

Cubic Polynomial

What are the Roots of a Linear Polynomial?

The value of a variable that makes the whole expression equal to zero is known as the root of the polynomial. In a linear polynomial, we always get only one root because it has only one variable with the highest power of 1. We can find the roots of a linear polynomial using the formula x = -b/a. Let us learn to find the roots of a linear polynomial using the following example:

Find the root of p(x) = 3x - 9.

Set the expression equal to 0:

3x - 9 = 0

Here, a = 3, b = -9

Use the formula to find the roots of a linear polynomial:

x = -(-9)/3

= 9/3 = 3

So, the root of the given linear polynomial is 3.

What is the Proof of Roots of Linear Polynomial?

To verify the roots of a linear polynomial using a formula, let’s take the general form of a linear polynomial: p(x) = ax + b, where a and b are real numbers and a cannot be 0. If a becomes 0, then the given expression cannot be a linear polynomial.

To find the values of x, we have to make the whole expression equal to 0.

So, ax + b = 0

Now, find the value of x:

ax + b = 0

ax = -b

x = -b/a

Hence, proved.

Linear Polynomial Functions

Linear polynomial functions can be represented as y = ax + b and are also known as first-degree polynomials. We know that polynomials might contain variables of different degrees, non-zero coefficients, positive exponents, and constant terms. A polynomial function can be represented in the form of a graph. The image given below shows the graph of different polynomial functions.

The linear polynomial function always forms a straight line in the graph and is represented as y = ax + b.

The graph of a quadratic polynomial is a curve, and it is also known as a parabola. It can be represented as y = ax² + bx + c.

The cubic polynomial takes the shape that is shown on the right side of the image, and it can be represented as y = ax³ + bx² + cx + d.

How to Solve Linear Polynomial Function?

We are trying to find the value of the variable that makes the whole expression equal to zero when we are solving a linear polynomial. This value is called the zero or root of a polynomial. Follow the steps below to solve a linear polynomial function:

Step 1: Write down the given polynomial

Step 2: Set the polynomial equal to zero.

Step 3: Solve the equation step by step to find the value of the variable x or y.

Example: Consider the polynomial f(x) = 5x + 10

Step 1: Set the polynomial to 0.

5x + 10 = 0

Step 2: Solve the equation

Subtract 10 from both sides
5x = -10

Divide both sides of the equation by 5.
x = -2

Therefore, the zero or root of the function is -2.

What is Zero of Linear Polynomial?

A linear polynomial is in the form of p(x) = ax + b, where a ≠ 0. It is easy to find the zero of a linear polynomial, as it has only one zero for this polynomial. To find the zero, we just set the expression equal to 0 and solve for x.

p(x) = ax + b

Set the expression to 0

ax + b = 0

Move the b to the other side, while moving b to the other side of the equation, its sign changes

ax = -b

Divide both sides by a:

x = -b/a

This is the value of x that makes the whole expression equal zero.
Note that if a = 0, then the x-term disappears, and it is not a linear polynomial anymore. It becomes just a number. 3x + 6 = 0, ½ x - 1 = 0 are some of the examples of linear polynomials.

Real Life Applications of Linear Polynomial

Linear polynomials are a simple but powerful tool that is used in many real-life situations and in various fields. Here is a detailed explanation of how linear polynomials are used.

Business and Economics: Businesses use linear polynomials to calculate the cost, revenue, and profit of the business. For example, to find out how many items a company must sell to cover costs, they use linear equations to compare the cost and income.

Engineering: In engineering, linear polynomials are used to design parts and to predict the result of stress vs strain models. Linear polynomials are used to simplify complex formulas and make quick predictions when there are small changes in a system.

Environmental Science: If pollution increases linearly every year, we can use linear polynomials to show how it changes over time.

Agriculture: In agriculture, farmers use linear polynomials to calculate the cost of seeds, fertilizers, and water depending on land area.

Common Mistakes and How To Avoid Them in Linear Polynomial

Solving linear polynomials is usually simple, but sometimes students make small mistakes that can lead to wrong answers. These mistakes may happen when moving terms, using the wrong signs, etc. Understanding these common errors is important because it helps to avoid confusion and improve problem-solving skills.

Mistake 1

Not changing the sign when moving a term

When moving a term from one side to the other side of the equation, we forget to change the sign and keep the sign the same on both sides, which leads to an incorrect calculation. Always change the signs when moving the terms.

For example, when simplifying 5x + 3 = 0, we write it as 5x = 3, which is incorrect, and it should be like 5x = -3.

Mistake 2

Thinking linear polynomials have more than one root

Sometimes we forget that linear equations have only one root and keep looking for another root in an expression. Always remember that a linear polynomial has only one root because the highest degree is 1.

Mistake 3

Thinking a = 0 is a linear equation

If p(x) = 0x + 5, it becomes 5. Since there is no variable left, it cannot be solved like an equation. Therefore, it is not a linear equation. Remember that a should not be 0 in a linear polynomial, or else it becomes a constant and not a linear equation.

Mistake 4

Writing the root without solving

Sometimes we guess the answer and write only the final answer, which can lead to a mistake. Show all steps and calculations before providing the final answer.