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Identify a rational expression.
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Identify opposite quantities within a rational expression.
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Detect cancelable factors within a rational expression and perform cancellations.
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Perform operations (add, subtract, multiply, divide) on rational functions.
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5.1: Simplify Rational Expressions
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Rational numbers are sometimes informally called fractions. Numbers such as 3/4 and −1/5 are rational numbers. When simplifying rational expressions, look for groups of variables or numbers that can be canceled to one. Cancel as many times as permits within a rational expression.
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5.2: Multiply Rational Expressions
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Multiplying rational expressions is very much the same as fraction multiplication in arithmetic. The numerators are multiplied to numerators. The denominators are multiplied to denominators. The common factors in the numerator and denominator are canceled before multiplying. That’s it! Two examples are shown below. Compare the similarities!
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5.3: Divide Rational Expressions
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Dividing rational expressions is very much the same as fraction division in arithmetic. The first step is to change the division to multiplication and take the reciprocal of the second fraction. A complex fraction is a fraction in which either the numerator is a fraction, or the denominator is a fraction, or both. To simplify complex fractions, translate the main fraction bar to division.
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5.4: Add and Subtract Rational Expressions
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A fraction is a proportion. The fraction communicates the number of parts out of the whole. For example, the egg carton pictured below contains 10 eggs. Part of the eggs are brown (7 eggs) while the rest are white (3 eggs). A fraction a/b quickly communicates the proportion of eggs that are brown or are white. Proportions can also be given as a decimal or percentage.