# Basic Properties of Rational Expressions

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1 Basic Properties of Rational Expressions A fraction is not defined when the denominator is zero! Examples: Simplify and use Mathematics Writing Style. a) x + 8 b) x 9 x 3 Solution: a) x + 8 (x + 4) x + 4 b) x 9 x 3 (x + 3)(x 3) x 3 x + 3

2 The Domain of a Rational Function: Examples: Find the domain of the following functions: a) f(x) x + 4 b) g(x) x + 5 x 4 c) h(x) x + 5 x 3 4x Solution: d) F(x) x x + 9 a) f(x) x + 4 This is a polynomial, and is defined for all values of x. So the domain is the set of real numbers. D R b) g(x) x + 5 x 4 This is a rational function and is not defined when the denominator is zero, or when x 4 0 x 4 So the domain is consists of all real numbers except that. c) h(x) x + 5 x 3 4x d) F(x) This is a rational function and is not defined when the denominator is zero, or when D {x R x 4 } x x + 9 This is a rational function and is not defined when the denominator is zero, or when x 3 4x 0 x x(x 4) 0 x(x )(x + ) 0 But this expression is never x 0,, So the domain is consists of all real numbers except these. zero for any real value of x. So the domain consists of all real numbers. D {x R x 0,, } D R

3 Multiplication and Division of Rational Expressions Example: Perform the indicated operation and simplify: x + x 3 x 3x 0 4x + x x x 9x 5 x x + x + x 3 x 3x 0 x x 4x + x 9x 5 x x + (x + 3)(x ) (x 5)(x + ) x(x ) (x + ) x(x + 3) (x + ) (x + )(x 5) (x )

4 Adding or Subtracting Fractions with Equal Denominators Examples: Perform the indicated operation and simplify: a) x x 4 + x 4 b) x x 4 4x x x 4 x + x 4 x (4x x ) x 4 x + (x + )(x ) x 4x + x x 4 x x 4x x 4 x(x ) (x + )(x ) x (x + )

5 Finding the LCD Step. Factor each denominator completely, including the prime factors of any constant factor. Step. Form the product of all the factors that appears in the complete factorizations. Step 3. The number of times any factors appears in the LCD is the most number of times it appears in any one factorization. Examples: Find the LCD for the given denominators: a) Denominators are 4, 30, and LCD b) Denominators are x 3 x and x 3 x x 3 x x (x ) x 3 x x(x ) x(x + )(x ) LCD x (x + )(x )

6 Adding or Subtracting Fractions with Unequal Denominators (FLEAS). Factor the rational expression.. Find the Least Common Denominator (LCD). 3. Equalize each denominators by replacing each fraction with an equivalent one whose denominator is the LCD. 4. Add or Subtract using RASFED. Example:

7 Examples: Perform the indicated operation and simplify. a) x x + x + b) x + x x + x x 4 (x + )(x ) (x + ) x + x x + x (x + )(x ) (x + ) (x + ) (x ) x (x + ) (x ) (x + ) (x + )(x ) x + x (x + )(x ) (x + ) (x ) (x + ) (x ) (x + 4x + 4) (x + x) (x + )(x ) x + x + (x + ) (x ) x + 4x + 4 x x (x + )(x ) x + 3 (x + ) (x ) x + 4 (x + )(x ) (x + ) (x + )(x ) (x )

8 Complex Fractions To simplify complex fractions: Step : Identify all fractions in the numerator and denominator and find the LCD. Step : Multiply the numerator and denominator by the LCD. Examples: 3 a) b) 7 xy x + y x + y x + y x + y xy (x + y) xy x + y xy (x + y) y + x

9 Example: Simplify the following: a) 9 y 3 y b) x + h x h y y 9 y 3 y 9y 3y y (3y + )(3y ) y(3y ) 3y + y x(x + h) x + h x x(x + h) h x (x + h) x(x + h) h x x h x(x + h) h h x(x + h) h x(x + h)

10 Long Division of Polynomials Monomial Denominator: When you divide a polynomial by a monomial, you must divide each term in the numerator by the denominator Examples: Perform the indicated operation. a) (x 3 6x + x) 3x b) Divide 5y 3 +0y 5y by 5y x3 6x + x 3x 5y 3 + 0y 5y 5y x3 3x 6x 3x + x 3x 5y 3 5y + 0y 5y 5y 5y x 3 x + 3 3y + 4y c) (6x 3 8x + 3x) x d) ( 6z 4 +6z 3 +8z +64z) 8z 6x3 8x + 3x x 6z4 +6z 3 +8z +64z 8z 6x3 x 8x 3x + x x 6z4 8z +6z3 8z + 8z 8z 64z 8z 8x 8x 3 + z 3 + z + z 8

11 Long Division of Polynomials Examples: Calculate the indicated quotients by long division: a) x3 x 7x + 3 x x 3 x 7x + 3 x + x 4x + + x + b) x4 8x 8 x x x 4 8x 8 x x + x + x 9 + x + 0 x x +

12 c) 6x4 + x 3 9x + 4 x + + 6x 4 + x 3 9x + 4 x 3x 3 + x + x Synthetic Division of Polynomials You can only use synthetic division when you divide a polynomial by a linear polynomial with linear coefficient. Examples: Calculate the indicated quotients by synthetic division: a) x3 x 7x + 3 x x 3 x 7x + 3 x + x 4x + + x +

13 b) x4 8x 8 x x 4 8x 8 x 3 x 3 + 3x + x x 3 c) x4 8 x x 4 8 x + 3 x 3 3x + 9x 7

14 Remainder Theorem When you divide a polynomial P(x) by the factor x c, the remainder is P(c). Thus we sometimes evaluate a polynomial P(x) when x c by performing the appropriate synthetic division. Examples : Let P(x) x 3 4x + 5. a) By direct substitution, evaluate P(). P(x) x 3 4x + 5 P() () 3 4() b) Find the remainder when P(x) is divided by x Remainder is 5 Examples : Let P(x) 4x 6 5x x 4 + 7x. Find P(4) P(4) 6 Note the problem is easier when we use the Remainder Theorem

15 Equations Involving Fractions To solve equations with (simple) fractions: Step : Identify all fractions in the equation and find the LCD. Step : Multiply the both sides of the equation by the LCD. Step 3: Solve the resulting equation. Step 4: Check the answer into the original problem. Examples: Solve the following: a) x 6 3 x 8 (x 6)(x 8) x 6 (x 6)(x 8) 3 x 8 (x 8) 3(x 6) x 6 3x 8 x x

16 b) z 4 z z z z 4 z 3 z z 4 z(z ) z z 4 z (z ) z (z ) z 4 z(z ) z (z ) z z 4 z (z ) z(z 4) z(z ) (z ) 4 z 4z z z z z 4z z 4z 0 0 The answer would be all real numbers, but we much check! all real numbers except 0, c) y y 3 y 9 y y 3 (y + 3)(y 3) (y + 3)(y 3) y y 3 (y + 3)(y 3) (y + 3)(y 3) (y + 3)(y ) (y + 3)(y 3) y + y 6 y 9 y 5

17 d) x 4 x + x x (x + )(x ) x + x(x ) x (x + )(x ) (x + )(x ) x (x + )(x ) x + x(x ) x (x + )(x ) + x(x + ) x x 4 + x + x x x + x 4 x x The only possible solution is x, but we must check! no solution

18 e) + 0 x + 3 x + 3 (x + )(x + 3) 0 + x + (x + )(x + 3) 3 x + 3 (x + )(x + 3) + 0(x + 3) 3(x + ) (x + 5x + 6) + 0x x + 6 x + 0x + + 0x x + 6 x + 7x (x + 9)(x + 4) 0 x 9/, 4 f) y + y y + 5 y + 6y + 3 y + y 6 (y + 3)(y ) y + y y + 5 y (y + 3)(y ) 6y (y+3)(y ) (y )(y + ) + (y + 3)(y + 5) (y + 3)(y ) + (6y + 3) y y + y + 8y + 5 y + y 6 + 6y + 3 y + 7y + 3 y + 7y + 7 y 4 y ± The solution would be y or y, but we must check! y

19 Example: It takes Rosa, traveling at 50 mph, 45 minutes longer to go a certain distance than it takes Maria traveling at 60 mph. Find the distance traveled. distance rate time Rosa x 50 x/50 Maria x 60 x/60 Important: 45 minutes 3/4 hours. We must use hours here! x 50 x x 50 x x 5x 5 x 5 5 miles

20 Example: Beth can travel 08 miles in the same length of time it takes Anna to travel 9 miles. If Beth s speed is 4 mph greater than Anna s, find both rates. Solution: distance rate time Beth 08 x /(x+4) Anna 9 x 9/x 08 x + 4 x(x + 4) 08 x x x(x + 4) 08 x x 9(x + 4) 08x 9x x 768 x 48 Beth 5 mph Anna 48 mph

21 Example: Toni needs 4 hours to complete the yard work. Her husband, Sonny, needs 6 hours to do the work. How long will the job take if they work together? Toni Sonny together 4 hours 6 hours x hours x x x x 3x + x 5x x 5 5 hours hours + 5 hours hours minutes hours 4 minutes

22 Example: Working together, Rick and Rod can clean the snow from the driveway in 0 minutes. It would have taken Rick, working alone, 36 minutes. How long would it have taken Rod alone? Rick 36 minutes Rod x minutes together 0 minutes 36 + x 0 80x 36 + x 80x 0 5x x 80 4x 45 x 45 minutes

23 Example: John, Ralph, and Denny, working together, can clean a store in 6 hours. Working alone, Ralph takes twice as long to clean the store as does John. Denny needs three times as long as does John. How long would it take each man working alone? John Ralph Denny together x hours x hours 3x hours 6 hours x + x + 3x 6 6x x + x + 3x 6x x x John minutes Ralph minutes Denny 33 minutes

24 Example: An inlet pipe on a swimming pool can be used to fill the pool in hours. The drain pipe can be used to empty the pool in 0 hours. If the pool is empty and the drain pipe is accidentally opened, how long will it take to fill the pool? inlet pipe drain pipe together hours 0 hours x hours 0 We subtract because the pipes are working against each other! 60x 0 x 60x x 5x 3x 60 x 60 x hours

25 Example: You can row, row, row your boat on a lake 5 miles per hour. On a river, it takes you the same time to row 5 miles downstream as it does to row 3 miles upstream. What is the speed of the river current in miles per hour? distance rate time downstream x 5/(5 + x) upstream 3 5 x 3/(5 x) x 3 5 x (5 + x)(5 x) x (5 + x)(5 x) 3 5 x 5(5 x) 3(5 + x) 5 5x 5 + 3x 0 8x 5 4 x 5 4 mph

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