GRE Rational Expressions: Key Concepts, Simplification, Practice Problems

Rational expressions (also called algebraic fractions) are ratios of two algebraic expressions containing unknown variables, wherein the numerator and denominator are polynomial expressions. To simplify a rational expression, the numerator and denominator are converted to their lowest forms. In GRE Math, rational expressions are a critical part of the algebra topic. Read on to explore the key concepts, simplification, and operations on rational expressions along with practice problems.

What is a Rational Expression?

Rational expressions are the ratio of two polynomials. In other words, a rational expression is a fraction with variables where both the numerator and denominator are polynomials. The denominator can never be zero since it would make the expression undefined. Rational expressions are expressed in the form of p(x) / q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0. Since rational expressions are similar to fractions, they follow the same rules for operations.

What is a polynomial?

Polynomials are mathematical expressions that combine numbers and variables (x, y) using addition, subtraction, and multiplication. A term is each part of a polynomial that is separated by a + or − symbol.

Example of a polynomial: x + 2 (This has 2 terms: x and 2)

To find the root or zero of a polynomial, set the entire expression equal to zero. But in case of rational expressions, only the numerator needs to be set equal to zero. This works only when the expression is fully simplified, meaning that the numerator and denominator have no common factors. For example, the fraction 3/9 is not in the lowest form. It can be reduced to ⅓ by taking 3 as a common factor. 

Similarly, the rational expression x² +2x/5x is not in its lowest form. By taking the common factor x from both the numerator and denominator, it can be simplified to (x+2)/5, which is the lowest form of this rational expression.

 How to Simplify a Rational Expression? 

Simplifying a rational expression involves reducing it to its simplest form by ensuring the numerator and denominator have no common factors other than 1. Similar to simplifying fractions, the primary step is identifying and cancelling out these common factors to achieve the reduced expression. The step-by-step procedure to simplify rational expressions is as given below:

• Identify and factorize both the numerator and denominator into their simplest polynomial forms.
• Identify the common factor between the numerator and denominator.
• Divide both the numerator and denominator by their common factors.
• Check for restrictions and note values that make the denominator zero. 
• Ensure that the expression is correct and fully simplified.

Example: Simplify x² -9/x² -3x

• Factor the numerator and denominator: (x+3)(x+3)/x(x-3).
• Cancel the common factor (x-3): x+3/3.
• Restriction: x ≠ 0 and x ≠ 3.

Operations on Rational Expressions

When performing operations on rational expressions, the process is similar to that used for fractions. Operations such as addition, subtraction, multiplication, and division can be applied to algebraic fractions the same way they can be applied to numeric fractions.

Adding and Subtraction of Rational Expressions

When adding or subtracting rational expressions, the denominators must be the same. If the denominators are already the same, you can simply add or subtract the numerators. If the denominators are different, you need to find the Least Common Denominator (LCD) to rewrite the expressions with matching denominators before performing the operation. The general formula for addition and subtraction of rational expressions is as given below:

• Addition: a/c + b/c = (a+b)/c
• Subtraction: a/c – b/c = (a – b)/c

To understand better, let’s take an example of a fraction first. 

Add and subtract ¾ and ⅚. 

Addition: ¾ + ⅚ = (9+10)/12 = 19/12

Subtraction: ¾ - ⅚ = (9-10)/12 = -1/12 

Here, we have normalised the denominators by finding the LCM of 4 and 6 (which is 12) and then performed the operations.

Let’s now apply the same method to perform the operations on rational expressions.

Adding Rational Expressions:

Question: Add 4/(x+3) and (x+5)/(x+3).

Solution: Since the denominators are the same:

4/(x+3) + (x+5)/(x+3) = 4+(x+5)/(x+3) = (x+9)/(x-3)

Final Answer: (x+9)/(x-3)

Subtracting Rational Expressions:

Question: Subtract 2/(x² -9) from 8/(x-3).

Solution: Since x²-9 = (x-3)(x+3): 8/(x-3) - 2/(x-3)(x+3)

The denominators are different, so find the LCM and rewrite the fractions:

8(x+3)/(x-3)(x+3) - 2/(x-3)(x+3)

Now, perform the operation:

8(x+3)-2/(x-3)(x+3) = 8x+24-2/x² -9 = 8x+22/x² -9

Final Answer: 8x+22/x² -9

Multiplication and Division of Rational Expressions

When multiplying two rational expressions, we multiply their numerators and denominators separately. However, we follow the standard principle of simplifying, factorizing, and cancelling common terms prior to performing the operation. The general formula for multiplying rational expressions is as given below:

Multiplication: a/b x c/d = ac/bd

Multiplying Rational Expressions:

Question: Multiply x+2/x-4 by x² -16/x+5.

Solution: Factor x² -16 = (x+4)(x-4):

x+2/x-4 X (x+4)(x-4)/x+5 = (x+2)(x+4)/(x-4)(x+5)

Cancel out the common factor x-4: (x+2)(x+4)/(x+5)

Final Answer: (x+2)(x+4)/(x+5)

To divide rational expressions, first flip the second fraction (reciprocal) and then multiply. Then follow the same procedure as the multiplication method. The general formula for dividing rational expressions is as given below:

Division: a/b ÷ c/d = a/b x d/c = ad/bc

Dividing Rational Expressions:

Question: Divide 5/x² -9 by 10/x-3

Solution: Rewrite as multiplication by the reciprocal:

5/x² -9 ÷ 10/x-3 = 5/x² -9 X x-3/10

Factor x² -9 = (x-3)(x+3): 5/(x-3)(x+3) X x-3/10

Cancel out x-3: 5/x+3 X 1/10 = 5/10(x+3)

Simplify: 1/2(x+3) 

Final Answer: 1/2(x+3) 

Restrictions of Rational Expressions

Restrictions are crucial to ensure the expression remains valid. Since rational expressions are fractions, the denominator cannot be equal to zero. To find the restrictions of rational expressions, set each factor of the denominator equal to zero and solve for the variable.

Example: In the rational expression x/x² -9, let’s factorize the denominator: x2-9 = (x+3) (x-3).

Now, let’s set each factor of the denominator equal to zero to find the restrictions.

x+3 = 0 -> x = -3
x -3 = 0 -> x = 3

Thus the restrictions are: x ≠ -3 and x ≠ 3.

Rational Expressions Calculator

A rational expressions calculator is an online tool designed to calculate the rationalized form of a given rational expression. It lets the user enter the numerator and denominator values and calculates the expressions in just a few seconds. A rational expression can be expressed as the quotient or fraction p/q or simply as the quotient of two polynomials.

Rational Expressions Practice Problems

Here are some rational expressions practice questions along with solutions for you to practice and enhance your skills.

Question 1: Simplify x2+5x+6 / x2+3x

Solution: Factor both numerator and denominator:
Numerator: x² +5x+6 = (x+2) (x+3) 
Denominator: x² +3x = x(x+3)

Cancel the common factor (x+3):
(x+2) (x+3) / x(x+3) = x+2 / x, x ≠ -3,0

Final Answer: x+2 / x. Restrictions: x ≠ -3,0 

Question 2: Add 1/x+1 + 1/x-1

Solution: 

Take the LCM: 1/x+1 = 1/x-1 = (x-1) + (x-1)/(x+1) (x-1)

Simplify (x-1)/(x+1) (x-1): 2x/x² -1

Final Answer: 2x/x² -1

Question 3: Subtract 2x/x² -4 - x/x+2

Solution: Factorize: x² -4 = (x+2) (x-2)

Take the LCM: 2x/(x+2) (x-2) - x/(x+2)

Multiply numerator and denominator of the second term by (x-2): 

2x/(x+2) (x-2) - x(x-2)/(x+2) (x-2)

Combine and simplify: 

2x-x² +2x/(x+2) (x-2) = -x² +4x/(x+2) (x-2) = x(-x+4)/(x+2) (x-2)

Final Answer: x(-x+4)/(x+2) (x-2)

Question 4: Multiply x² -9/x² -1 by x+1/x² +3x

Solution: 

Factorize all expressions:  x²-9 = (x-3) (x+3), x2-1 = (x-1) (x+1),x² 2+3x = x(x+3)

Rewrite the product: (x-3) (x+3)/(x-1) (x+1). x+1/x(x+3)

Cancel the common factors (x+3) and (x+1): x-3/x(x-1)

Final Answer: x-3/x(x-1)

Question 5: Divide x² -4x+4/x² -9 ÷ x-2/x+3

Factorize all expressions: x² -9 = (x-3) (x+3),  x²-1 = (x-1) (x+1), x² +3x = x(x+3)

Rewrite the product: (x-3) (x+3)/(x-1) (x+1) . x+1/x(x+3)

Cancel the common factors (x+3) and (x+1): x-3/x(x-1)

Final Answer: (x+1): x-3/x(x-1)