Define a rational expression: a quotient of two polynomials. ..( 3 10) (3 2) Rational expressions have the same properties as rational numbers:

Transcription

1 1 UNIT 7 RATIONAL EXPRESSIONS & EQUATIONS Simplifying Rational Epressions Define a rational epression: a quotient of two polynomials. A rational epression always indicates division EX: 10 means..( 10) ( ) Rational epressions have the same properties as rational numbers: 1) 5, where... 0 a ), where.. b b A rational epression is in its simplest form when the numerator and denominator have no common factors ecept 1 or -1. Simplest Form a a To simplify rational epressions: Not in Simplest Form 5( 4) ( 4) 1. factor the numerator and the denominator. common factors will simplify to 1. We will restrict the value of the variables so the values will not make the denominator zero ( ) 5 5 The two 5 s in the second step simplify to 1 y y y y y y y y y ( )( 1) 1 ( 1)( 1) 1 The two (y+1) s simplify to 1 Try This. Simplify. a. 1y y

2 b. + + c. a 1 a a 1 ADDITIVE INVERSES 8 4 What is the additive inverse? (8 4 ) 8 4 Eample 4 1( 4) 1(4 ) 1 4 (4 ) (4 ) You FACTORED a -1 from the numerator to make the factor within the parentheses the same as the denominator and you used the commutative property of addition to change the order so it looked EXACTLY like the denominator. Eamples 1. 6 ( ) ( ) 1( ). 1 y (1 y)(1 + y) 1(y 1)(1 + y) 1(1 + y) y 4y + (y 1)(y ) (y 1)(y ) (y ) 1 y y

3 Try This. Simplify. b. b 7 7 b c. 5 a (a 5) d MULTIPLYING RATIONAL EXPRESSIONS Multiplying rational epressions is just like multiplying fractions Numerator times numerator and denominator times denominator ( ) ( ) a a ( a )( a ) a ( a) NOT a 4 a 9 STEPS FOR MULTIPLYING RATIONAL EXPRESSIONS 1. Completely factor the numerators and denominators. Look for cross canceling of factors (items WITHIN parentheses) and simplfy. Multiply the remaining factors. Writing them as the product of the factors and not completely multiplied. Do NOT FOIL. 4. Always look to Simplify Completely. 1 ( 1) y 6 y 5 ( y ) y 5 ( y )( y 5) ( y )( y 5)

4 4 Try This. Multiply. Simplify the product. a. 4a 4 6 5a b ( + ) + 4 c. 4m + 4 m 1 DIVIDING RATIONAL EXPRESSIONS STEPS To Dividing Rational Epressions Copy Dot Flip Factor the numerators and denominators. Look to cross cancel and simplify Multiply across Always simplify completely Eample ( 1) ( 1)( 1) ( 1) ( 1) ( 1) ( 1) ( 1)( 1) ( 1) ( 1) ( 1) ( 1) ( 1) Eample 1 8n 5 n 9 8n5 9 n 7n5 6n 1n

5 5 Eample Eample ( + 8)(9) ( + 4)(9) ( + 4) ( + 4) ( + 1)( + ) + 1 ( + )( + 1) + + Try This a. 9n 7 9n 14 b. 4m 8 5 m 10 c. 4 ADDITION LIKE DENOMINATORS Remember that when adding or subtracting fractions with common denominators, the operation sign only applies to the numerators. Complete the operation and then simplify if possible. Eample1. Add and simplify 4m + 5m 4m + 5m 9m m

6 6 Eample 6a a + + 4a a + 6a + 4a 10a a + a + Eample ( + )( 5) + 1 SUBTRACTION LIKE DENOMINATORS The additive inverse/distributing the negative but only applied to the numerator. When subtracting one rational epression from another: 1. Replace the subtraction sign with addition. Replace the second fraction with its opposite (adding the opposite), remembering to distribute the negative to every term in the second epression. Then add Eample 4. Subtract and simplify a a + a 4 a (a 4) a + 4 (a + ) a + a + a + (a + ) Try this d. 4m + 5 m 1 + m 1 m 1 e. y + 4y y + y y 1 y +

7 7 ADDITION & SUBTRACTION WITH UNLIKE DENOMINATORS Steps 1. Find a LCM and change to common denominators. Combine like terms. Factor 4. Simplify (1 ) Find the LCM Steps 1. Factor. Take the largest eponent of the common terms.. Multiply the remaining part in the end. 8 y and 1y and... 1

8 8 and y... and... y y y and y y and...4 Eample ( + ) + 5( + ) 4( + ) ( + ) ( + ) + 8 0( + ) 1 0( + ) Eample 6 k k 5 5 k k k 5 1(k 5) k + k 5 k k 5 1(k 5) 1 1 k k 5 (k 5)

9 9 Try this. Add or subtract and simplify f. + y y y g. 4 b + b b 6 SOLVING RATIONAL EQUATIONS A rational equation is an equation containing one or more rational epressions. Here are some eamples: Steps To solve rational equations. 1. Find the LCM of all the denominators.. Multiply both sides of the equation by the LCM.

10 10. Cross Cancel and Multiply 4. Solve for the variable. 5. Check Eample (4 ) (4 ) Check: Eample ( + 6 ) This is a second degree equation. Thus we set the equation equal to zero ( + )( + ) or + 0 or

11 11 Check: Eample ( 1) ( 1) 1 1 ( 1)( + 1) or or 1 Check: In eample the number 1is an Etraneous solution (an etra solution obtained by multiplying the equation by a variable). CHECK the solutions back into the original equation. We will see that 1 cannot be a solution because it makes the denominator 0. Try This a. + 1 b.

12 c COMPLEX RATIONAL EXPRESSIONS A comple rational epression has a rational epression in its numerator, denominator, or both. Eamples: 1 in the numerator y 1 both y both To simplify a comple rational epression, multiply the numerator and denominator by an epression equivalent to 1. Use the LCM of the denominators. Eample 1.

13 The LCM is eight and 8 8 is equivalent to Use the Distributive Property to multiply everything in the numerator by eight as well as everything in the denominator. After all the multiplying, cross canceling, simplifying, and this is the final simplified answer. Eample The LCM is , so multiply the numerator and denominator by the LCM. 1 1 Simplify the numerator and denominator ( 1) 1 ( 1)( 1) 1 Remember to then look for factoring to simplify the rational epression.

14 14 Eample (A different option with copy dot flip) 4 y y Try this a y y 4 y y + y 4( y) ( y ) 4( y) ( y)( + y) b c. ( + )( 1) +