Rational Expression

The fractions that have variables in the numerator, the denominator, or both are called rational expressions. The rational expression has the form p(x)/q(x), where q(x) ≠ 0, and p(x) and q(x) are polynomials. If either the numerator or the denominator is not a polynomial, then it is not considered as a rational expression. x + 1/2x + 2,  2x2 + 2x/5x + 1 are some examples of rational expressions. 
 

Rational expressions are used to reduce the expression into the simplest form by removing the common factors from the numerator and denominator. Follow the steps given below for simplifying the rational expressions:

Step 1: Factorize the numerator and denominator.

Step 2: Eliminate common factors from both the numerator and denominator.

Step 3: Write the remaining expressions after simplifying, that is the answer.  

Step 4: Add a restriction by identifying values that make the denominator zero.
Example: x2 - 9 x - 3
Factorize the numerator: x2 - 9 = (x + 3)(x - 3)
Cancel the common factors: (x + 3)(x - 3) x - 3 = x + 3
x ≠ 3, because if we apply it to the original equation, the denominator becomes zero. 
 

Roots are the values of x that make the expression to become 0. Follow the steps given below for finding the roots of rational expressions. 

Step 1: Set the numerator equal to 0, because a fraction is 0 when the numerator is 0.

Step 2: Solve the equation to find the value of x.

Step 3: Make sure that the denominator is not 0, because dividing by 0 is not allowed.

Example: Find the roots of x - 5 x + 2
Make the numerator to 0, x - 5 = 0
x = 5
Check the denominator: x + 2  = 0, x = -2
x= -2 is not allowed, but x = 5 won’t make the denominator 0.
Therefore, the root is 5.
 

Rational expression can also be added, subtracted, multiplied, and divided like regular fractions. The operations in rational expressions are:

• Addition
• Subtraction
• Multiplication
• Division
   

Adding and subtracting rational expressions is similar to working with numeral fractions. We can follow the steps given below for adding and subtracting rational expressions.

Step 1: Find a common denominator.

Step 2: Express each rational expression with the same common denominator.

Step 3: Add and subtract the numerator

Step 4: Simplify the expression

Example: Simplify 1/x + 2/x + 1
The common denominator is x(x + 1)
Rewrite both equations,
(x + 1)/x(x + 1) + 2x/x(x + 1)
Add the numerators,
x + 1 +2x/x(x + 1) = 3x + 1/x(x + 1)
The final answer is: 3x + 1/x(x + 1)
 

For multiplying rational expressions, we need to multiply the numerator and the denominator together. By using the following steps we can multiply the rational expressions

Step 1: Multiply the numerators

Step 2: Multiply the denominators

Step 3: Simplify the expression by removing the common factors. 

Example: Simplify: 2/x × 3/x + 1
Multiply:  2 ×3/x(x + 1) =  6/x(x + 1)
The final answer is: 6/x(x + 1)
 

Dividing rational expression is similar to dividing regular fractions. For dividing the rational expressions, follow the steps given below:

Step 1: Flip the second fraction to find its reciprocal. 

Step 2: Multiply the terms.

Step 3: Simplify the fractions if you can.

Example: Simplify: 4/x × 2/x + 1
Take the reciprocal of the second fraction,
4/x × x + 1/2
Multiply: 4(x + 1)/2x
Simplify, 2(x + 1/)x
Therefore, the final answer is 2(x + 1/)x
 

Rational expression follows specific rules that help in simplifying, solving, and performing operations like addition, subtraction, multiplication, and division. There are four basic rules of rational expressions, they are:

Rule 1: When multiplying expressions with the same base, we can add the exponents.
xm × xn = xm + n
For example, x2 × x3 = x2 + 3 = x5

Rule 2: We can subtract the powers of two expressions when we divide two expressions with the same base.
xrxs = xr - s
Example: x5x3 = x5 - 3 = x2

Rule 3: When an exponent is raised to another power in an expression, we can multiply the exponents.
(xu)v = xu × v
For example: (x3)2 = x3 × 2 = x6

Rule 4: If an exponential expression has a negative exponent, the result is the reciprocal of the expression with a positive exponent.
x-h = 1xh
Example: x-10 = 1x10
 

Rational expressions have unique graphical and algebraic features that help describe their behavior. These characteristics describe how the expression behaves at certain points when the input value becomes very large or very small. The four main characteristics of rational expressions are:

• Zeroes
• Holes
• Vertical Asymptote
• Horizontal Asymptote

Zeros: Zeros are the values of x that cause the numerator to become zero.
f(x) = x - 3x + 2
Set the numerator 0 
x - 3 = 0, x = 3
So, x = 3 is a zero of an expression.

Hole: When a common factor is present in both the numerator and the denominator, a hole occurs. In a graph, the hole makes a gap there.
f(x) = (x - 1)(x + 2)(x - 1)(x + 5) 
Since the factor (x - 1) appears in both numerator and denominator, there is a hole at x = 1. 

Vertical Asymptote: It is a vertical line where the expression is undefined due to a non-cancelled factor in the denominator. 
f(x) = 1x - 4
Make the denominator as 0.
x - 4 = 0, x = 4
Therefore, x = 4 is a vertical asymptote.

Horizontal Asymptote: It is a flat line that the graph gets close to, far to the left or right. The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. 
f(x) = 2x2 + 1x2 + 5
Degree of numerator = 2
Degree of denominator = 2
So, the horizontal asymptote is, y = 21 = 2
 

When we divide a number by zero, we won’t get any result; it is not allowed in math. In the same way, if the denominator of a rational expression becomes zero, the whole expression does not make sense. 
Example: x/ x + 5 
Here, if we put x = -5, the denominator becomes 0. So, we say that x = -5 is not allowed and it is called restriction. 
To find the restrictions, take the denominator and set it to 0 and solve it. Any value that makes the denominator zero is called a restricted value. 
 

Rational expressions play an important role in solving practical problems across various fields. They are commonly used in many real-life situations, and some of their real-world applications are listed below.

• Engineering: In engineering, rational expressions are used to calculate how fast things work or move. If two machines are doing the same job at the same time, we use the rational expression to find out how fast they work together. 

• Medicine: Doctors use rational expressions to decide how much medicine to give to the patient. The dosage of the medicine depends on the person’s weight. 

• Chemistry: In chemistry, rational expressions are used to calculate concentrations when mixing chemicals. If we mix two salt solutions, we can use rational expressions to find the concentration of the final mix.

• Travel: Pilots and drivers use rational expressions to plan the speed, fuel, and time. A pilot finds the average by dividing the total distance by the total time.