Rational expressions

Excluded values

A rational expression is a fraction of polynomials. An excluded value makes the denominator 0. Find them by setting the denominator equal to 0, before you cancel anything.

Example: \(\dfrac{x + 1}{x^2 - 9}\): \(x^2 - 9 = 0\), so \(x \ne 3\) and \(x \ne -3\).

Simplifying

Factor the top and bottom completely, then cancel common factors.

Example: \(\dfrac{x^2 - 9}{x^2 + 5x + 6} = \dfrac{(x + 3)(x - 3)}{(x + 3)(x + 2)} = \dfrac{x - 3}{x + 2}\), \(\;x \ne -3, -2\).

Watch out: cancel factors, never terms: \(\dfrac{x + 5}{5} \ne x\).

Opposite factors

\(\dfrac{a - b}{b - a} = -1\), because \(b - a = -(a - b)\).

Example: \(\dfrac{4 - x}{x - 4} = -1\); \(\;\dfrac{2x - 6}{3 - x} = \dfrac{2(x - 3)}{-(x - 3)} = -2\).

Multiplying

Factor everything, cancel common factors (top with bottom, across both fractions), then multiply what is left.

Example: \(\dfrac{x^2 - 4}{3x} \cdot \dfrac{6x^2}{x + 2} = \dfrac{(x + 2)(x - 2) \cdot 6x^2}{3x(x + 2)} = 2x(x - 2)\).

Dividing

Flip the second fraction (use its reciprocal) and multiply.

Example: \(\dfrac{x + 3}{x - 1} \div \dfrac{x^2 - 9}{x^2 - 1} = \dfrac{x + 3}{x - 1} \cdot \dfrac{(x - 1)(x + 1)}{(x + 3)(x - 3)} = \dfrac{x + 1}{x - 3}\).

Adding and subtracting: same denominator

Combine the numerators over the denominator. Put the subtracted numerator in parentheses. Simplify at the end.

Example: \(\dfrac{3x}{x - 2} - \dfrac{x + 4}{x - 2} = \dfrac{3x - (x + 4)}{x - 2} = \dfrac{2x - 4}{x - 2} = 2\).

Watch out: without the parentheses you get \(3x - x + 4\), which is wrong.

Adding and subtracting: the LCD

Factor each denominator. The LCD uses each factor the greatest number of times it appears. Rewrite each fraction over the LCD, then combine.

Example: \(\dfrac{2}{x} + \dfrac{3}{x + 1} = \dfrac{2(x + 1) + 3x}{x(x + 1)} = \dfrac{5x + 2}{x(x + 1)}\).

Example: \(\dfrac{x}{x^2 - 4} - \dfrac{1}{x - 2} = \dfrac{x - (x + 2)}{(x + 2)(x - 2)} = \dfrac{-2}{(x + 2)(x - 2)}\).

Complex fractions

A complex fraction has fractions inside it. Multiply its top and bottom by the LCD of all the small fractions, then simplify.

Example: \(\dfrac{1 + \frac{1}{x}}{1 - \frac{1}{x^2}}\), times \(x^2\) on top and bottom: \(\dfrac{x^2 + x}{x^2 - 1} = \dfrac{x(x + 1)}{(x + 1)(x - 1)} = \dfrac{x}{x - 1}\).

Work problems ("together")

If one person takes \(a\) hours alone and another takes \(b\) hours, together they take \(t\) hours where \(\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{t}\).

Example: 3 hours and 6 hours: \(\dfrac{1}{3} + \dfrac{1}{6} = \dfrac{1}{2}\), so together it takes 2 hours.

Distance-rate problems

Time = distance ÷ rate. When the times are equal, set the two time fractions equal.

Example: A boat goes 24 miles downstream in the same time it goes 16 miles upstream; the current is 2 mph. Let \(r\) be the boat's speed in still water: \(\dfrac{24}{r + 2} = \dfrac{16}{r - 2}\) gives \(24r - 48 = 16r + 32\), so \(r = 10\) mph.