8-4 Rational Expressions

Objectives

To simplify rational expressions

To multiply and divide rational expressions

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The expression 1 + 1 x 1 plus , 1 over x  in the Solve It is equivalent to the rational expression x + 1 x . fraction x plus 1 , over x end fraction . .  A rational expression is the quotient of two polynomials. You will find that, at different times, it is helpful to think of rational expressions as ratios, as fractions, or as quotients.

Essential Understanding You can use much of what you know about multiplying and dividing fractions to multiply and divide rational expressions.

A rational expression is in simplest form when its numerator and denominator are polynomials that have no common divisors.

You simplify a rational expression by dividing out the common factors in the numerator and the denominator. Factoring the numerator and denominator will help you find the common divisors.

A rational expression and any simplified form must have the same domain in order to be equivalent.

x 2 − x − 6 x 2 + x − 2 = ( x − 3 ) ( x + 2 ) ( x − 1 ) ( x + 2 )  and  x − 3 x − 1 , x ≠ − 2 , fraction x squared , minus x minus 6 , over x squared , plus x minus 2 end fraction . equals . fraction open , x minus 3 , close . open , x plus 2 , close , over open , x minus 1 , close . open , x plus 2 , close end fraction . and . fraction x minus 3 , over x minus 1 end fraction . comma x not equal to negative 2 comma  are equivalent.

In the example above, you must exclude − 2 negative 2  from the domain of x − 3 x − 1 fraction x minus 3 , over x minus 1 end fraction  because − 2 negative 2  is not in the domain of x 2 − x − 6 x 2 + x − 2 . fraction x squared , minus x minus 6 , over x squared , plus x minus 2 end fraction . .  Note that this restriction is not evident from the simplified expression x − 3 x − 1 . fraction x minus 3 , over x minus 1 end fraction . .

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