Section 2.1: Reduce Rational Expressions

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1 CHAPTER Section.: Reduce Rational Expressions Section.: Reduce Rational Expressions Ojective: Reduce rational expressions y dividing out common factors. A rational expression is a quotient of polynomials. Examples of rational expressions include: x x x x 0 and x and a a and DETERMINING EXCLUDED VALUES FOR A RATIONAL EXPRESSION It is important to rememer that the denominator of a fraction cannot have a value of zero. A rational expression is undefined when its denominator equals 0. W must exclude all value(s) of the variale that make a denominator zero. We must determine the value(s) that will give us zero in the denominator, and then exclude those values as possile values of the variale. Example. State the excluded value(s). 4x Find values of x for which denominator is 0 x Set the denominator equal to 0 x 0 Solve the equation x Excluded value: x We can evaluate the rational expression for any other value of x except for. Example. State the excluded value(s). x 0 Excluded values: x 0 and x x Find values of x for which denominator is 0 Set the denominator equal to 0 x 0 Factor the left side x() 0 Set each factor equal to zero or 0 Solve each equation x Second equation solved Page

2 CHAPTER Section.: Reduce Rational Expressions EVALUATING A RATIONAL EXPRESSION We evaluate rational expressions y sustituting the given value for the variale and then simplifying using the order of operations. Example. Evaluate the expression for the given value of the variale. x 4 x 6x8 when x 6 Sustitute 6 in for each variale ( 6) 4 ( 6) 6( 6) ( 6) Exponents first Multiply Add and sutract Reduce 8 4 SIMPLIFYING A RATIONAL EXPRESSION Just as we reduced the fraction in the previous example, often a rational expression can e reduced. When we reduce, we divide out common factors. A rational expression is reduced, or simplified, if the numerator and denominator have no common factors other than or. If the rational expression only has monomials, we can reduce the coefficients, and divide out common factors of the variales. Example 4. Simplify the expression. xy 4 xy 6 4 y Reduce and y dividing out the common factor of ; 4 Sutract the exponents: x 6 4 x and y y ; 4 Negative exponents move to denominator ecause y 4 y Page 4

3 CHAPTER Section.: Reduce Rational Expressions If the rational expression has more than one term in either the numerator or denominator, we need to first factor the numerator or denominator and then divide out common factors. Example. Simplify the expression. 8 8x 6 Denominator has a common factor of 8 ; factor. 8 Reduce y dividing 8 and 8 y 4. 8( x ) 7 ( x ) Example 6. Simplify the expression. x 8x 6 ( ) 6( ) Numerator has a common factor of ; denominator has a common factor of 6 ; factor oth the numerator and denominator Divide out common factor of ( ), and divide oth and 6 y Example 7. Simplify the expression. x x 8x ( x ) ( x ) ( x ) ( x ) Factor oth the numerator and the denominator; numerator is the difference of two squares; denominator can e factored using the ac method; Divide out common factor of ( x ) x x It is important to rememer that we cannot divide terms, only factors. This means if there are any or signs etween the parts we want to reduce, we cannot reduce. In the previous x example, we had the solution we cannot divide out the x s ecause they are terms x (separated y or ), not factors (separated y multiplication). Page

4 CHAPTER Section.: Reduce Rational Expressions FACTORS THAT ARE OPPOSITES Sometimes, factors in the numerator and denominator are opposites. To simplify, factor from the numerator or denominator. Example 8. Simplify the expression. x x x x ( x ) x Rewrite the numerator Factor from the numerator Divide out common factor of ( x ) Example. Simplify the expression. 7 x x 4 x 7 x 4 Rewrite the numerator Factor from the numerator; Factor denominator as difference of two squares ( x 7) ( x 7 ) ( x 7) Divide out common factor of ( x 7) x 7 Page 6

5 CHAPTER Section.: Reduce Rational Expressions Practice Exercises Section.: Reduce Rational Expressions Evaluate. ) 4v 6 when v 4 a 4) a a when a ) when ) 4 4 when 0 x ) x 4x when x 4 6) n n 6 n when n 4 State the excluded value(s). 7) k 0k k 0 8) 7 p 8p 6 p 0x 6 ) 6x 0 r r ) r 0 n ) 0n x 0 0) 8x 80x 4) ) 6n 6n n n 4 0m 8m ) 0m 0v 0v 6) v v Simplify each expression. x 7) 8x 4a 8) 40a ) 0) 4 8x 0x 0x The Practice Exercises are continued on the next page. Page 7

6 CHAPTER Section.: Reduce Rational Expressions Practice Exercises: Section. (continued) Simplify each expression. ) ) 8m p x ) x 4) x x ) n n 8 6) x 7 7 x x 7) x 8x 7 8) ) 0) ) 8m 6 8x 8x 4r 6 6r n n ) 4n 7n v 4 ) v 4v 60 0x 0 4) 0x 40 x ) 0x 4x 4x 6) k k k 64 7) 6 a 0 0a 4 p 8 8) p 4p 4 n n 0 ) n 0 x 40 40) x 0x 80 x 4) x 4 r 6r 4) r 40r x 0x 8 4) 7x ) 0 0 7n n 6 4) 4n 6 v 46) v 7 n n 47) 6n 6 6x 48 48) 4x 6x 7a 4) 6a 4k 0) k 6a 4 4a 0 k k 8k k Page 8

7 CHAPTER Section.: Reduce Rational Expressions ANSWERS to Practice Exercises Section.: Reduce Rational Expressions ) 4) undefined ) ) ) 6) 6 7) 0 8) 0, ) 0) 0, 0 ) 0 ) 0 ) 4) 0, ) 8,4 6) 0, 7 7) 7 x 6 ) 4 x 8) a 0) x The Answers to Practice Exercises are continued on the next page. Page

8 CHAPTER Section.: Reduce Rational Expressions ANSWERS to Practice Exercises: Section. (continued) ) m 4 0 ) ) 4) ) 0 p 6) cannot e simplified 7) x 7 8) 7 m ) 8x 7( x ) 0) 7 r 8 8r ) n 6 n 6 ) 7 ) v 0 ( x ) 4) x 4 ) x 7 x 7 6) k 4 k 8 7) a a 8) p ) n 40) x ( x ) 4) ( x ) r 4) ( r ) 4) ( x 4) 4 44) 8 4) 7 n 4 4 ( v ) 46) v 47) 48) 4) ( n ) 6( n ) 7x 6 (4)( x) 7a (a ) 0) (k ) ( k ) Page 60