Rational Functions

Introduction

Rational expressions are algebraic fractions containing polynomials in their numerators and denominators. Just as we can add, subtract, multiply, and divide numerical fractions, we can perform the same four operations with rational expressions.

For example, consider the numerical fractions:

1/3 + 1/6 = 1/2

The same principles apply when variables are involved:

1/x + 2/x = 3/x, where x ≠ 0.

However, operations with rational expressions require careful attention to factoring, common denominators, domain restrictions, and simplification.

These skills are important in algebra, calculus, physics, engineering, and economics. Rational expressions appear in problems involving rates, work, speed, electrical circuits, and other mathematical relationships.

In this lesson, we will learn how to add, subtract, multiply, and divide rational expressions, identify restrictions, and simplify final answers completely.

Understanding Rational Expressions

A rational expression is an expression that can be written as:

where P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial.

Examples include:

• (x + 2)/(x − 3)

• 5/x

• (x² − 4)/(x + 2)

• (2x² + 3x − 1)/(x² − 9)

Before performing any operation, remember:

The denominator of a rational expression cannot equal zero.

For example:

is undefined when x = 5.

Therefore, x ≠ 5 is a restriction.

When combining rational expressions, we must preserve all restrictions from the original expressions. Division also requires that the expression being divided by is not zero.

The Four Operations: Overview

The least common denominator (LCD) is the smallest polynomial expression, up to a nonzero constant factor, that contains all the factors needed to form a common denominator.

Adding Rational Expressions

Adding Expressions with the Same Denominator

When two rational expressions have the same denominator, add their numerators and keep the denominator unchanged.

The general rule is:

where c ≠ 0.

Worked Example 1: Adding with a Common Denominator

Simplify:

Step 1: Identify restrictions.

x − 4 = 0

x = 4

Therefore, x ≠ 4.

Step 2: Combine the numerators.

Step 3: Collect like terms.

3x + 2 + x + 5 = 4x + 7

Final answer:

No common factors can be cancelled, so the expression is fully simplified.

Adding Expressions with Different Denominators

When denominators are different, we must first find a common denominator.

For numerical fractions:

1/3 + 1/4 = 4/12 + 3/12 = 7/12

The same method works for algebraic fractions.

Worked Example 2: Adding with Different Linear Denominators

Simplify:

Step 1: Identify restrictions.

x ≠ 0 and x ≠ −1.

Step 2: Find the LCD.

The denominators are x and x + 1.

LCD = x(x + 1)

Step 3: Rewrite each fraction.

Step 4: Add the numerators.

Step 5: Expand and simplify.

2x + 2 + 3x = 5x + 2

Final answer:

Worked Example 3: Adding with Factored Denominators

Simplify:

Step 1: Factor the denominators.

x² − 4 = (x − 2)(x + 2)

Step 2: Identify restrictions.

x ≠ 2 and x ≠ −2.

Step 3: Find the LCD.

LCD = (x − 2)(x + 2)

Step 4: Rewrite the second fraction.

Step 5: Combine numerators.

1 + 2(x − 2)

= 1 + 2x − 4

= 2x − 3

Final answer:

Subtracting Rational Expressions

Subtraction follows the same principles as addition.

When denominators are equal:

where c ≠ 0.

The important difference is that subtraction requires careful use of negative signs.

Worked Example 4: Subtracting with the Same Denominator

Simplify:

Step 1: Identify restrictions.

x ≠ −2.

Step 2: Combine numerators.

Step 3: Distribute the negative sign.

5x + 3 − 2x + 4

= 3x + 7

Final answer:

Worked Example 5: Subtracting with Different Denominators

Simplify:

Step 1: Identify restrictions.

x ≠ 1 and x ≠ −2.

Step 2: Find the LCD.

LCD = (x − 1)(x + 2)

Step 3: Rewrite.

Step 4: Combine numerators.

3(x + 2) − 2(x − 1)

Step 5: Expand.

3x + 6 − 2x + 2

= x + 8

Final answer:

Finding the Least Common Denominator

For more complicated expressions, finding the LCD is often the most important step.

Consider:

Factor the denominators:

x² − 9 = (x − 3)(x + 3)

x² + 3x = x(x + 3)

The LCD must contain every distinct factor, using the highest power needed.

Therefore:

LCD = x(x − 3)(x + 3)

The restrictions are:

x ≠ −3, x ≠ 0, and x ≠ 3.

Rewrite:

Combine:

Simplify:

Factor the numerator:

There are no common factors to cancel.

Final answer:

3(x − 2)/[x(x − 3)(x + 3)], where x ≠ −3, 0, 3.

Multiplying Rational Expressions

Multiplication is often simpler than addition or subtraction because a common denominator is not required.

The general rule is:

where b ≠ 0 and d ≠ 0.

The most efficient approach is usually to factor before multiplying, then cancel common factors.

Worked Example 6: Multiplying Simple Rational Expressions

Simplify:

Step 1: Identify restrictions.

The original denominators are x + 2 and 6.

Therefore, x ≠ −2.

Step 2: Multiply the fractions.

Step 3: Cancel the common factor (x + 2).

3x/6

Step 4: Simplify numerical coefficients.

x/2

Final answer: x/2, where x ≠ −2.

Worked Example 7: Multiplying with Quadratic Factors

Simplify:

Step 1: Identify restrictions.

x + 4 ≠ 0 and x − 3 ≠ 0.

Therefore:

x ≠ −4 and x ≠ 3.

Step 2: Factor the numerator.

x² − 9 = (x − 3)(x + 3)

Step 3: Rewrite.

Step 4: Cancel common factors.

Cancel (x − 3) and (x + 4).

The remaining expression is:

x + 3

Final answer: x + 3, where x ≠ −4 and x ≠ 3.

Worked Example 8: Multiplying Two Quadratic Rational Expressions

Simplify:

Step 1: Identify restrictions.

x² − 4 = (x − 2)(x + 2)

The other denominator is x + 3.

Therefore:

x ≠ 2, x ≠ −2, and x ≠ −3.

Step 2: Factor completely.

x² + 5x + 6 = (x + 2)(x + 3)

x² − 4 = (x − 2)(x + 2)

Rewrite:

Step 3: Cancel common factors.

Cancel (x + 2), (x + 3), and (x − 2).

All factors cancel, leaving:

1

Final answer: 1, where x ≠ −3, −2, and 2.

Even though the final expression is a constant, the original restrictions must remain.

Dividing Rational Expressions

Division of rational expressions follows the same rule as division of numerical fractions.

To divide by a fraction, multiply by its reciprocal.

For example:

The same rule applies to algebraic fractions.

where b, c, and d must all be nonzero.

There is an additional restriction when dividing rational expressions: the divisor itself cannot equal zero.

Worked Example 9: Dividing Simple Rational Expressions

Simplify:

Step 1: Identify restrictions.

The denominators require x ≠ 0.

The divisor (x + 2)/3 must not equal zero.

Therefore, x ≠ −2.

Step 2: Multiply by the reciprocal.

Step 3: Cancel (x + 2).

3/x

Final answer: 3/x, where x ≠ 0 and x ≠ −2.

Worked Example 10: Dividing with Factoring

Simplify:

Step 1: Identify restrictions.

The original denominators require:

x ≠ −3.

The divisor must also be nonzero:

(x − 4)/(x + 3) ≠ 0

Therefore:

x ≠ 4.

Step 2: Multiply by the reciprocal.

Step 3: Factor the difference of squares.

x² − 16 = (x − 4)(x + 4)

Rewrite:

Step 4: Cancel common factors.

The remaining expression is:

x + 4

Final answer: x + 4, where x ≠ −3 and x ≠ 4.

Worked Example 11: Dividing Quadratic Rational Expressions

Simplify:

Step 1: Identify restrictions.

The original denominators are:

x² − 9 = (x − 3)(x + 3)

and x − 3.

Therefore:

x ≠ −3 and x ≠ 3.

The divisor (x + 2)/(x − 3) must not equal zero.

Therefore, x ≠ −2.

Step 2: Rewrite using the reciprocal.

Step 3: Factor completely.

Step 4: Cancel common factors.

Cancel (x + 2), (x + 3), and (x − 3).

The result is:

1

Final answer: 1, where x ≠ −3, −2, and 3.

Mixed Operations with Rational Expressions

Some problems involve more than one operation.

The correct order of operations is important:

1. Simplify expressions inside brackets.

2. Perform multiplication and division from left to right.

3. Perform addition and subtraction from left to right.

4. Factor and simplify the final result.

5. Preserve all original restrictions.

Worked Example 12: Combining Addition and Subtraction

Simplify:

Step 1: Identify restrictions.

x ≠ 2 and x ≠ 0.

Step 2: Combine fractions with the same denominator.

3/(x − 2) − 1/(x − 2) = 2/(x − 2)

The expression becomes:

2/(x − 2) + 4/x

Step 3: Find the LCD.

LCD = x(x − 2)

Step 4: Rewrite.

Step 5: Combine numerators.

2x + 4(x − 2)

= 2x + 4x − 8

= 6x − 8

Step 6: Factor the numerator.

6x − 8 = 2(3x − 4)

Final answer:

Worked Example 13: Combining Multiplication and Addition

Simplify:

Step 1: Identify restrictions.

x ≠ −1.

Step 2: Perform multiplication first.

The expression becomes:

x/2 + 3/2

Step 3: Add.

(x + 3)/2

Final answer: (x + 3)/2, where x ≠ −1.

Verifying Rational Expression Operations

A reliable way to check an answer is to substitute a permitted numerical value into both the original and simplified expressions.

Consider:

Choose x = 2.

Original:

2/2 + 3/3 = 1 + 1 = 2

Simplified:

(5(2) + 2)/[2(3)] = 12/6 = 2

Both expressions give the same value.

This supports the simplification.

However, numerical substitution alone is not a complete proof. Algebraic manipulation establishes equivalence across the permitted domain.

Real-World Applications

Application 1: Combining Work Rates

Suppose one machine can complete a job in 4 hours and another machine can complete the same job in 6 hours.

The first machine completes 1/4 of the job per hour.

The second completes 1/6 of the job per hour.

Their combined work rate is:

Find the LCD:

LCD = 12

Rewrite:

3/12 + 2/12 = 5/12

Together, the machines complete 5/12 of the job each hour.

The time needed to complete one job is:

1 ÷ (5/12) = 12/5 = 2.4 hours.

Therefore, the machines working together can complete the job in 2 hours and 24 minutes.

Application 2: Electrical Resistance

For two resistors connected in parallel:

Suppose:

R₁ = 6 Ω

R₂ = 3 Ω

Then:

1/R = 1/6 + 1/3

Find the LCD:

1/R = 1/6 + 2/6 = 3/6 = 1/2

Taking the reciprocal gives:

R = 2 Ω

The equivalent resistance is 2 Ω.

This demonstrates how adding rational expressions is used in electrical engineering.

Application 3: Average Speed

A vehicle travels a distance d at speed v₁ and returns over the same distance at speed v₂.

Total distance:

2d

Total time:

d/v₁ + d/v₂

Average speed:

Combine the denominator:

Therefore:

Cancel d:

For speeds of 60 km/h and 90 km/h:

Average speed = 2(60)(90)/(60 + 90)

= 10,800/150

= 72 km/h.

Notice that the average speed is not simply the arithmetic mean of the two speeds, because the vehicle spends different amounts of time travelling at each speed.

Common Mistakes

Mistake 1: Adding Denominators

Incorrect:

1/x + 1/(x + 2) = 2/(2x + 2)

Correct:

Use the LCD x(x + 2).

Mistake 2: Forgetting Negative Signs During Subtraction

Incorrect:

(3x + 2) − (x − 4) = 2x − 2

Correct:

3x + 2 − x + 4 = 2x + 6

Always distribute the negative sign across every term in the second numerator.

Mistake 3: Finding an LCD for Multiplication

A common denominator is not required when multiplying rational expressions.

Instead, multiply numerators and denominators, preferably after factoring and cancelling common factors.

Mistake 4: Forgetting to Use the Reciprocal

Incorrect:

(a/b) ÷ (c/d) = ac/bd

Correct:

(a/b) ÷ (c/d) = ad/bc

Mistake 5: Cancelling Terms Instead of Factors

Incorrect:

(x + 3)/(x + 5) = 3/5

Correct:

The expression cannot be simplified by cancellation because the x terms are part of sums.

Mistake 6: Forgetting Original Restrictions

Even when a denominator factor cancels, its zero must remain excluded.

For division, also exclude any value that makes the original divisor zero.

Mistake 7: Leaving an Answer Partially Simplified

For example:

(6x + 12)/(3x)

can be simplified by factoring the numerator:

6(x + 2)/(3x)

Then reducing 6/3 gives:

2(x + 2)/x, where x ≠ 0.

Always check whether additional common factors remain.

Key Terms

• Rational Expression: A fraction whose numerator and denominator are polynomials.

• Numerator: The expression above a fraction bar.

• Denominator: The expression below a fraction bar.

• Common Denominator: A denominator shared by two or more fractions.

• Least Common Denominator (LCD): The least polynomial common multiple of the denominators, up to a nonzero constant.

• Factor: An expression multiplied by another expression to form a product.

• Reciprocal: The multiplicative inverse of a nonzero number or expression.

• Equivalent Expressions: Expressions that produce the same output for every input in a specified common domain.

• Domain Restriction: A condition excluding inputs that make an expression undefined.

• Excluded Value: A value that makes an original denominator zero or, in a division problem, makes the divisor zero.

• Simplification: Rewriting an expression in an equivalent, less complicated form.

• Cancellation: Dividing numerator and denominator by a shared nonzero factor.

• Complex Fraction: A fraction containing one or more fractions in its numerator or denominator.

• Work Rate: The amount of work completed per unit of time.

Key Takeaways

1. Addition and subtraction of rational expressions require a common denominator.

2. When denominators are equal, combine the numerators and keep the denominator.

3. The LCD is found by factoring denominators and including every required factor at its highest power.

4. Multiplication involves multiplying numerators together and denominators together.

5. Division requires multiplying by the reciprocal of the divisor.

6. Factoring before multiplication or division often makes simplification easier.

7. Only common factors can be cancelled, not individual terms.

8. Restrictions from original denominators must always be preserved.

9. In division, the original divisor must also be nonzero.

10. Final answers should be factored or reduced wherever possible and checked for correctness.

Check Your Understanding

Question 1: Addition with a Common Denominator

Simplify:

Question 2: Addition with Different Denominators

Simplify:

Question 3: Subtraction

Simplify:

Question 4: Multiplication

Simplify:

Question 5: Division

Simplify:

Question 6: Adding with Factored Denominators

Simplify:

Question 7: Mixed Operations

Simplify:

Question 8: Real-World Application

One pump fills a tank in 5 hours. Another pump fills the same tank in 10 hours.

a) Write the work rate of each pump.

b) Calculate their combined work rate.

c) Determine how long the pumps take to fill the tank when working together.

Fully Worked Answers

Answer 1

Given:

(2x + 3)/(x − 4) + (x + 5)/(x − 4)

Combine numerators:

(2x + 3 + x + 5)/(x − 4)

= (3x + 8)/(x − 4)

Final answer: (3x + 8)/(x − 4), where x ≠ 4.

Answer 2

Given:

1/x + 2/(x + 3)

LCD = x(x + 3)

Rewrite:

(x + 3)/[x(x + 3)] + 2x/[x(x + 3)]

Combine:

(x + 3 + 2x)/[x(x + 3)]

= (3x + 3)/[x(x + 3)]

Factor:

Final answer: 3(x + 1)/[x(x + 3)], where x ≠ 0 and x ≠ −3.

Answer 3

Given:

5/(x + 1) − 2/(x − 2)

LCD = (x + 1)(x − 2)

Rewrite:

Expand:

5x − 10 − 2x − 2

= 3x − 12

Factor:

3(x − 4)

Final answer: 3(x − 4)/[(x + 1)(x − 2)], where x ≠ −1 and x ≠ 2.

Answer 4

Given:

(x² − 9)/(x + 2) × (x + 2)/(x − 3)

Factor:

Cancel common factors:

x + 3

Final answer: x + 3, where x ≠ −2 and x ≠ 3.

Answer 5

Given:

(x + 4)/x ÷ (x + 4)/2

Multiply by the reciprocal:

Cancel (x + 4):

2/x

The original denominator requires x ≠ 0.

The divisor must be nonzero, so x ≠ −4.

Final answer: 2/x, where x ≠ 0 and x ≠ −4.

Answer 6

Given:

1/(x² − 4) + 3/(x + 2)

Factor:

x² − 4 = (x − 2)(x + 2)

LCD = (x − 2)(x + 2)

Rewrite:

Combine:

[1 + 3(x − 2)]/[(x − 2)(x + 2)]

Expand:

1 + 3x − 6 = 3x − 5

Final answer: (3x − 5)/[(x − 2)(x + 2)], where x ≠ −2 and x ≠ 2.

Answer 7

Given:

2/x + 3/(x − 1) − 1/x

Combine like denominators:

2/x − 1/x = 1/x

Now:

1/x + 3/(x − 1)

LCD = x(x − 1)

Rewrite:

Combine:

(x − 1 + 3x)/[x(x − 1)]

= (4x − 1)/[x(x − 1)]

Final answer: (4x − 1)/[x(x − 1)], where x ≠ 0 and x ≠ 1.

Answer 8

a) Work rates

First pump:

1/5 tank per hour

Second pump:

1/10 tank per hour

b) Combined work rate

1/5 + 1/10

= 2/10 + 1/10

= 3/10 tank per hour

c) Time required

Time = 1 ÷ (3/10)

= 10/3 hours

= 3⅓ hours

= 3 hours and 20 minutes.

This completes Operations with Rational Expressions, covering addition, subtraction, multiplication, division, complete simplification, restrictions, and practical applications.