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9.1: Simplify Rational Expressions
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To simplify a rational expression, factor both the numerator and the denominator, and remove common factors from both the numerator and the denominator. A simplified rational expression has only one division, and a single numerator and denominator. If the expressions cannot be factored, then the rational expression cannot be simplified.
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9.2: Multiply Rational Expressions
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To multiply rational expressions, multiply the numerator expressions and multiply the denominator expressions. Then, if possible, simplify by factoring both the numerator and denominator and removing common factors. Try to use factor by grouping when working with a 4-term polynomial. Note that common denominators are not needed when multiplying rational expressions!
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9.3: Add and Subtract Rational Expressions
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To add or subtract rational expressions, think of this as fractions with variables. A common denominator (called the LCD) is needed for addition and subtraction. To find the LCD, first fully factor all denominators. Build the LCD from the factors found in all denominators. Multiply each factor the greatest number of times it occurs in either expression. If the same factor occurs more than once in both expressions, multiply the factor the greatest number of times it occurs in either expression.
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9.4: Rationalize Algebraic Fractions
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If the denominator of a rational expression contains sums or differences involving radicals, it is good form to always rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator contains the same terms, but opposite operations (addition or subtraction).