Mathematics.

expressions

Rational Expressions

Algebra I30 minDifficulty3 out of 10

You should know: factoring

Overview

A rational expression is a ratio of two polynomials, such as (x+3)/(x²-9), analogous to how a rational NUMBER is a ratio of two integers. Rational expressions appear whenever a quantity is described as one polynomial divided by another — rates, densities, and averages. Because the denominator is a polynomial, values that make it zero must be excluded from the domain, and simplifying a rational expression almost always begins with factoring both numerator and denominator.

Intuition

Treat a rational expression exactly like a fraction of numbers: to simplify, factor the top and bottom and cancel common factors; to add or subtract, find a common denominator; to multiply, multiply straight across (after factoring and canceling); to divide, flip and multiply. The one new danger compared to numeric fractions is that the denominator is now an expression in x, so you must explicitly exclude any x-value that would make it zero.

Interactive Graph

A rational expression with a vertical asymptote

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Formal Definition

Definition

A rational expression has the general form:

P(x)Q(x),Q(x)≢0\frac{P(x)}{Q(x)}, \quad Q(x) \not\equiv 0

P and Q are polynomials; the expression is undefined wherever Q(x) = 0

General form

Properties

Simplifying

P(x)Q(x)=A(x)B(x) after canceling common factors of P,Q\frac{P(x)}{Q(x)} = \frac{A(x)}{B(x)} \text{ after canceling common factors of } P, Q

Domain restriction

x is excluded whenever Q(x)=0x \text{ is excluded whenever } Q(x)=0

Multiplication

ab⋅cd=acbd\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}

Division

ab÷cd=ab⋅dc\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}

Applications

Combined resistance of resistors in parallel, 1/R = 1/R₁ + 1/R₂, is solved using rational expression arithmetic.

Worked Examples

  1. Factor the numerator (difference of squares) and denominator (trinomial).

    (x−3)(x+3)(x+2)(x+3)\frac{(x-3)(x+3)}{(x+2)(x+3)}

  2. Cancel the common factor (x+3), noting x ≠ -3.

    =x−3x+2,x≠−3,−2=\frac{x-3}{x+2}, \quad x\neq -3,-2

Answer: (x-3)/(x+2), with x ≠ -3, -2

Practice Problems

Difficulty 3/10

Simplify (x² - 4x)/(x² - 16), stating the excluded values.

Difficulty 4/10

Divide: (x+3)/(x-2) ÷ (x²-9)/(x-2).

Common Mistakes

Common Mistake

Canceling individual terms instead of common FACTORS, e.g. simplifying (x+3)/(x+5) by canceling the x's or the constants.

Only common FACTORS (things being multiplied) can cancel between numerator and denominator, never individual terms within a sum. (x+3)/(x+5) cannot be simplified further at all.

Common Mistake

Forgetting to state domain restrictions after simplifying, since the restriction from the original (unsimplified) denominator can 'disappear' after canceling.

The excluded values are determined by the ORIGINAL denominator before simplifying, not the simplified one — a canceled factor still represents a value that made the original expression undefined.

Summary

  • A rational expression is a ratio of two polynomials, P(x)/Q(x).
  • Simplify by factoring numerator and denominator, then canceling common factors — never individual terms.
  • Values that make the ORIGINAL denominator zero must always be excluded from the domain.
  • Multiply/divide/add/combine rational expressions using the same rules as numeric fractions.

References

  1. WebsiteWikipedia — Rational function