Simplifying and operating on rational expressions
A rational expression is a fraction of polynomials. Working with them follows exactly the same rules as ordinary fractions, with factoring doing the heavy lifting.
Rational expressions
A rational expression is a quotient of polynomials, such as \(\frac{x + 2}{x^2 - 9}\). It is undefined wherever the denominator is zero, here at x = 3 and x = -3, so those values are excluded from its domain. Always note excluded values before simplifying.
Simplifying
Factor numerator and denominator, then cancel common factors: \(\frac{x^2 - 4}{x^2 + 5x + 6} = \frac{(x + 2)(x - 2)}{(x + 2)(x + 3)} = \frac{x - 2}{x + 3}\), with \(x \neq -2, -3\). Only factors cancel, never terms: in \(\frac{x + 5}{x}\), the x's cannot be cancelled.
Opposite factors
Factors like \(x - 5\) and \(5 - x\) are opposites: \(5 - x = -(x - 5)\). So \(\frac{5 - x}{x - 5} = -1\) for \(x \neq 5\). Spotting opposites avoids missed cancellations.
Multiplying and dividing
Multiply by factoring everything, cancelling, then multiplying what remains: \(\frac{x^2 - 9}{2x} \cdot \frac{4x^2}{x + 3} = \frac{(x + 3)(x - 3)}{2x} \cdot \frac{4x^2}{x + 3} = 2x(x - 3)\). To divide, multiply by the reciprocal of the divisor.
Least common denominator
To add or subtract, rewrite each fraction over a least common denominator (LCD), the product of every factor to its highest power. The LCD of \(\frac{3}{x}\) and \(\frac{5}{x + 2}\) is \(x(x + 2)\), giving \(\frac{3(x + 2) + 5x}{x(x + 2)} = \frac{8x + 6}{x(x + 2)}\).
Subtracting carefully
When subtracting, the minus sign applies to the whole numerator: \(\frac{2x}{x - 1} - \frac{x + 3}{x - 1} = \frac{2x - x - 3}{x - 1} = \frac{x - 3}{x - 1}\). Forgetting to distribute the minus is the most common error.
Complex fractions
A complex fraction has fractions within its numerator or denominator, such as \(\frac{1 + \frac{1}{x}}{1 - \frac{1}{x^2}}\). Multiply top and bottom by the LCD of the inner fractions, \(x^2\): \(\frac{x^2 + x}{x^2 - 1} = \frac{x(x + 1)}{(x + 1)(x - 1)} = \frac{x}{x - 1}\).
Worked example
Two pipes fill a tank: one alone in a hours, the other in b hours. Together they fill \(\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}\) of the tank per hour, so the combined time is \(\frac{ab}{a + b}\) hours. With a = 3 and b = 6, that is \(\frac{18}{9} = 2\) hours.
Key takeaways
Rational expressions are fractions of polynomials, undefined where the denominator is zero; note excluded values first. Simplify by factoring and cancelling common factors, never terms, and watch for opposites like x - 5 and 5 - x. Multiply by factoring and cancelling; divide by multiplying by the reciprocal. Add and subtract over the least common denominator, distributing any minus sign across the whole numerator. Simplify complex fractions by multiplying top and bottom by the LCD of the inner fractions.
Flashcard
Quotient of polynomials
Check your understanding
Simplify \(\frac{x^2 - 4}{x^2 + 5x + 6}\).