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

The root of a polynomial is the value of x that makes the whole expression equal to zero. In a linear polynomial, we always get only one root because the highest power of the variable is 1.

To find the root, we use this simple rule:

\(x = -\frac{b}{a}\)


Here, the linear polynomial looks like this:


\(p(x) = ax + b\)

 

For 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.

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 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.

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.

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. 

Learning linear polynomials can be fun and easy when children connect equations to real-life examples. These tips and tricks help parents guide their child to understand, visualize, and practice linear polynomials effectively.


 

• Relate y = ax + b to everyday situations like cost, distance, or savings.
  
   

• Think “y = slope × x + start” to recall slope (rate of change) and y-intercept (starting point).
  
   

• Plot points from a table; a straight line confirms it’s linear.
  
   

• Identify variables, constants, and rates first, then write the equation step by step.
  
   

• Create a table of x and y values to make graphing easier and spotting patterns faster.

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

• Robotics: Robots use linear polynomials to calculate linear motion paths, speed, and position. For example, moving an arm at a constant rate or predicting distance over time uses a linear equation.
  
   
• Engineering: In engineering, engineers use linear polynomials to plan and measure things that change at a steady rate. For example, when designing a ramp, they can use the equation\( h = mx + b\), where m is how steep the ramp is, x is the distance along the ramp, and b is the starting height. This helps them make sure the ramp is safe and works correctly.
   
• Computer Animation & Graphics: In animation, linear polynomials help move objects at a constant speed or create smooth transitions. They are used in coding positions, scaling, and motion over time.
  
   
• Physics: Linear polynomials model relationships with constant rates, such as distance = speed × time + initial position, or Hooke’s Law for springs in the elastic region.
  
   

• Business and Economics: In business and economics, linear polynomials help to calculate costs, revenue, and profit.