7.2 Reducing Rational Functions

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628 Chapter 7 Rational Functions At this point, 2x − 2x 3 3x 3 + 4x 2 − 3x − 4 = 2x(1 + x)(1 − x) (x + 1)(x − 1)(3x + 4) . Because we have 1 − x in the numerator and x − 1 in the denominator, we will factor out a −1 from 1 − x, and because the order of factors does not affect their product, we will move the −1 out to the front of the numerator. 2x − 2x 3 2x(1 + x)(−1)(x − 1) −2x(1 + x)(x − 1) 3x 3 + 4x 2 = = − 3x − 4 (x + 1)(x − 1)(3x + 4) (x + 1)(x − 1)(3x + 4) We can now cancel common factors. 2x − 2x 3 −2x(1 + x)(x − 1) 3x 3 + 4x 2 = − 3x − 4 (x + 1)(x − 1)(3x + 4) −2x(1 + x)(x − 1) = (x + 1)(x − 1)(3x + 4) = −2x 3x + 4 Note that x + 1 is identical to 1 + x and cancels. Thus, 2x − 2x 3 3x 3 + 4x 2 − 3x − 4 = −2x 3x + 4 for all values of x, provided x ≠ −1, 1, or −4/3. These are the restrictions, values of x that make denominators equal to zero. (22) The Sign Change Rule for Fractions Let’s look at an alternative approach to the last example. First, let’s share the precept that every fraction has three signs, one on the numerator, one on the denominator, and a third on the fraction bar. Thus, −2 3 has understood signs Let’s state the sign change rule for fractions. + −2 +3 . The Sign Change Rule for Fractions. Every fraction has three signs, one on the numerator, one on the denominator, and one on the fraction bar. If you don’t see an explicit sign, then a plus sign is understood. If you negate any two of these parts, • numerator and denominator, or • numerator and fraction bar, or • fraction bar and denominator, then the fraction remains unchanged. Version: Fall 2007
Section 7.2 Reducing Rational Functions 629 For example, let’s start with −2/3, then do two negations: numerator and fraction bar. Then, + −2 +3 = −+2 +3 , or with understood plus signs, −2 3 = −2 3 . This is a familiar result, as negative two divided by a positive three equals a negative two-thirds. On another note, we might decide to negate numerator and denominator. Then −2/3 becomes + −2 +3 = +2 −3 , or with understood plus signs, −2 3 = 2 −3 . Again, a familiar result. Certainly, negative two divided by positive three is the same as positive two divided by negative three. They both equal minus two-thirds. So there you have it. Negate any two parts of a fraction and it remains unchanged. On the surface, this seems a trivial remark, but it can be put to good use when reducing rational expressions. Suppose, for example, that we take the original rational expression from Example 20 and negate the numerator and fraction bar. 2x − 2x 3 3x 3 + 4x 2 − 3x − 4 = − 2x 3 − 2x 3x 3 + 4x 2 − 3x − 4 Note how we’ve made two sign changes. We’ve negated the fraction bar, we’ve negated the numerator (−(2x − 2x 3 ) = 2x 3 − 2x), and left the denominator alone. Therefore, the fraction is unchanged according to our sign change rule. Now, factor and cancel common factors (we leave the steps for our readers — they’re similar to those we took in Example 20). 2x − 2x 3 3x 3 + 4x 2 − 3x − 4 = − 2x 3 − 2x 3x 3 + 4x 2 − 3x − 4 2x(x + 1)(x − 1) = − (x + 1)(x − 1)(3x + 4) 2x(x + 1)(x − 1) = − (x + 1)(x − 1)(3x + 4) = − 2x 3x + 4 But does this answer match the answer in equation (22) It does, as can be seen by making two negations, fraction bar and numerator. − 2x 3x + 4 = −2x 3x + 4 Version: Fall 2007
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7.2 Reducing Rational Functions

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