Rational Expressions
Grade 11 · Math · Study Guide
A rational expression is a ratio of two polynomials. Master simplification, operations, and finding excluded values.
A rational expression has the form P(x)/Q(x), where P and Q are polynomials and Q(x) ≠ 0. Excluded values are any x that make the denominator zero; always identify them before simplifying.
To simplify, factor numerator and denominator fully and cancel common factors. When multiplying, multiply numerators and denominators then simplify. When dividing, multiply by the reciprocal of the divisor.
Adding or subtracting rational expressions requires a common denominator, just like numeric fractions. Find the least common denominator (LCD) by taking the LCD of the polynomial factors.
| Operation | Rule |
|---|---|
| Multiply | Multiply numerators; multiply denominators; simplify. |
| Divide | Multiply by reciprocal of divisor. |
| Add/Subtract | Find LCD; rewrite each fraction; combine numerators. |
| Simplify | Factor fully; cancel common factors. |
| Pattern | Factored Form |
|---|---|
| Difference of squares | a² − b² = (a+b)(a−b) |
| Trinomial (leading 1) | x² + bx + c = (x + p)(x + q), where p+q=b, pq=c |
| GCF | Factor out the greatest common factor first |
Key terms
- Rational expression
- A fraction P(x)/Q(x) where P and Q are polynomials and Q ≠ 0.
- Excluded value
- Any value of x that makes the denominator zero; the expression is undefined there.
- Least common denominator (LCD)
- The smallest expression divisible by each denominator; required when adding or subtracting rational expressions.
- Complex fraction
- A fraction that contains fractions in its numerator, denominator, or both.
Practice questions
What is the excluded value of (x + 3)/(x − 5)?
- x = 3
- x = −3
- x = 5
- x = −5
Show answer
The denominator x − 5 = 0 when x = 5, making the expression undefined.
Simplify: (x² − 9)/(x + 3).
Show answer
x² − 9 = (x+3)(x−3). Cancel (x+3) to get x − 3.
What is (2/x) · (x²/6) simplified?
- x/3
- x²/3
- 2x/6
- 12/x
Show answer
(2/x) · (x²/6) = 2x²/(6x) = x/3.
Add: 1/(x) + 2/(x+1). Express as a single fraction.
Show answer
LCD = x(x+1). Rewrite: (x+1)/(x(x+1)) + 2x/(x(x+1)) = (3x+1)/(x(x+1)).
Which correctly divides (4/x²) ÷ (2/x)?
- 2/x
- 8/x³
- 2x
- 4/2x
Show answer
(4/x²) · (x/2) = 4x/(2x²) = 2/x.
Fun facts
- Rational expressions are used in physics to model combined resistance in parallel circuits: 1/R_total = 1/R₁ + 1/R₂.
- Any proper rational function can be decomposed into simpler fractions — a technique called partial fractions used heavily in calculus.
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