Introduction to Rational Expressions

Learn what rational expressions are, how to identify them, and understand their key properties including domain restrictions.

Advanced25 minLesson
Printable worksheet

Definition

A rational expression is a fraction where both the numerator and denominator are polynomials.
Rational Expression=polynomialpolynomial\text{Rational Expression} = \frac{\text{polynomial}}{\text{polynomial}}
Just like fractions with numbers, the denominator cannot equal zero.
Examples of rational expressions:
x+3x−2x2−4x+152x−63x2+2x−1x2+5\frac{x + 3}{x - 2} \quad \frac{x^2 - 4}{x + 1} \quad \frac{5}{2x - 6} \quad \frac{3x^2 + 2x - 1}{x^2 + 5}
Key concept - Domain Restrictions: When the denominator equals zero, the expression is undefined. The values that make the denominator zero are called excluded values or restrictions.

Try it now

Which of the following is a rational expression?

Worked Examples

Which of these are rational expressions? a) x2+3xx−1\frac{x^2 + 3x}{x - 1} b) xx+2\frac{\sqrt{x}}{x + 2} c) 5x3−8\frac{5}{x^3 - 8}
1

Check expression (a)

Numerator: x2+3xx^2 + 3x (polynomial). Denominator: x−1x - 1 (polynomial) → YES - rational expression
2

Check expression (b)

Numerator: x\sqrt{x} (NOT a polynomial - has a radical) → NO - not a rational expression
3

Check expression (c)

Numerator: 55 (constant polynomial). Denominator: x3−8x^3 - 8 (polynomial) → YES - rational expression

Common Mistakes

Setting the numerator equal to zero instead of the denominator

Why it's wrong: When finding restrictions, we need to find when the expression is undefined. Division by zero is undefined, not division into zero.

Correct: Always set the DENOMINATOR equal to zero to find restrictions. When the numerator is zero, the expression simply equals zero.

Forgetting to factor the denominator completely

Why it's wrong: A quadratic denominator like x2−9x^2 - 9 has TWO roots, so there are TWO restrictions.

Correct: Always factor the denominator completely before solving. x2−9=(x+3)(x−3)x^2 - 9 = (x + 3)(x - 3) gives restrictions at x=−3x = -3 AND x=3x = 3.

Thinking 05\frac{0}{5} and 50\frac{5}{0} are the same

Why it's wrong: 05=0\frac{0}{5} = 0 (zero divided by anything is zero), but 50\frac{5}{0} is undefined (cannot divide by zero).

Correct: Zero in the numerator gives zero. Zero in the denominator is undefined.

Interactive Visual

Function graph

-8-6-4-202468-4-2024xyx = -3x = 3

Tap or hover over the graph to read off values.

y = (x + 5)/(x² - 9)Asymptote

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which of the following is a rational expression?

Why It Matters

Rational expressions appear throughout advanced mathematics and real-world applications:
  • Physics: The formula for lens magnification is hiho=−dido\frac{h_i}{h_o} = -\frac{d_i}{d_o}
  • Chemistry: Concentration calculations use ratios of polynomials
  • Economics: Cost per unit is often expressed as total costquantity\frac{\text{total cost}}{\text{quantity}}
  • Engineering: Electrical resistance in parallel circuits uses 1RT=1R1+1R2\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}
Understanding rational expressions is essential for calculus, where limits of rational functions are a core concept.

Real World Applications

Average Speed Problems

When traveling different distances at different speeds, the average speed is a rational expression.

Example:

If you drive 100 km at speed vv and return at speed v+20v + 20, the average speed for the whole trip is 2×100100v+100v+20\frac{2 \times 100}{\frac{100}{v} + \frac{100}{v+20}}

Work Rate Problems

When two people work together, their combined rate involves rational expressions.

Example:

If Alice completes a job in xx hours and Bob in yy hours, working together they complete it in xyx+y\frac{xy}{x + y} hours.

Electrical Circuits

Parallel resistance formulas use rational expressions.

Example:

For two resistors in parallel: Rtotal=R1×R2R1+R2R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}

Key Takeaways

  • 1A rational expression is a fraction with polynomials in both numerator and denominator
  • 2The denominator can NEVER equal zero (division by zero is undefined)
  • 3To find domain restrictions, set the denominator equal to zero and solve
  • 4Factor the denominator completely to find ALL restrictions
  • 5Always check for restrictions before evaluating a rational expression

Frequently Asked Questions

The term 'rational' comes from 'ratio.' A rational expression is a ratio (fraction) of two polynomials, just like a rational number is a ratio of two integers.
The term 'rational' comes from 'ratio.' A rational expression is a ratio (fraction) of two polynomials, just like a rational number is a ratio of two integers.
Division asks 'how many times does the divisor fit into the dividend?' Zero fits into any number infinitely many times, making the answer undefined. There's no number that, when multiplied by zero, gives a non-zero result.
Yes! The number 55 is a constant polynomial (degree 0), and xx is a polynomial (degree 1). So 5x\frac{5}{x} is a rational expression with restriction x≠0x \neq 0.

Glossary

Rational expression
A fraction where both the numerator and denominator are polynomials
Domain restriction
A value of the variable that makes the expression undefined (when denominator equals zero)
Excluded value
Another term for domain restriction; a value that must be excluded from the domain
Undefined
When an expression has no valid numerical value, typically due to division by zero
Today's challengeOne problem, three tries

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