FLC Ch 6. Simplify. State any restrictions (if necessary). a) b) Simplify each. List all restrictions on the domain. Next, graph the function f.

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1 Math 0 Intermediate Algebra Defn Sec 6.: Rational Expressions and Functions: Multiplying and Dividing A polynomial divided by another nonzero polynomial is called a rational expression. If P and Q are polynomials in x with Q 0, then r(x) = P is called a rational function. Q Ex Simplify. State any restrictions (if necessary). a) b) 0yz 4 3y + 0y y z 9 3y 4y 5 Ex g(x) = Simplify each. List all restrictions on the domain. Next, graph the function f. 8x 6 x 4 f(x) = ( x + ) (x ) x Ex 3 Simplify. State any restrictions (if necessary). a) b) c) y + 0y + 5 y y + 3y 9 y + 5 a 3 b 3 3a + 9ab + 6b a + ab + b a b 4a a 4 a a Page of 4

2 d) (As time permits) 5x 5y 7x 3 + 8y 3 x xy + y 9x 6xy + 4y Ans: (x + y)/(x y)(x 3y) x y, y/3 6x + 4y 0x 5y Good Exercises: f(x) = (x + )3 g(x) = 3 x + 4 h(x) = (x + 3 ) Do problems from y = x Handout Ex 4 Intro to Sec 6.: Rational Expressions and Functions: Adding and Subtracting Find the LCD. 3 5 x and x 56(x ) (x + ) and 4(x )(x + )(x + ) x, x + x + y, x, and y 4x, 4x 8 9x, x + 4 Sec 6.3: Complex Rational Expressions *Method : LCD Method - Avoid Multiplying by Reciprocal *generally preferred Method : Multiply by Reciprocal EC?: Determine which method is more efficient. Ex 5 Simplify using each method. Method TWO Method ONE Page of 4

3 Ex 6 Simplify. a) b) 3xy x y yx x x x x 5 x + 8x + x 5x 4 c) d) PP y y 3 y y 3 y 7 y 6 x 5 x 5 4 x x + 5 e) f) In-Class Prob: Ans: y 7 x x x y y 3y y + 5y + 4 3y y y y + 3y 4 y+ Page 3 of 4

4 g) DO h) PP x x 7x x 4 x x 0 0 5x + 0 Ans: 5x 5x Complete Problems from Function Worksheet IV (Graphs Review A) Sec 6.6: Division of Polynomials Type I (easier): Type II (more difficult): Dividing by a monomial. Break up numerator. Dividing by a binomial (or polynomial with more than term). Use Long Division (LD) OR Synthetic Division (SD) (sec 6.7) Ex 7 Divide and check. a) b) Fractions 5x 3 + 0x 3x + 5 0x (3x 3 5x 3x ) (3x ) c) (3x 4 + x 3 x x + 5) (x ) Page 4 of 4

5 d) e) x 3 x + 6 x 3 8 x + x *answer only f) Practice Problem x 4 + 4x + 6 x + Answer: x x + g) (As time permits) Find a simplified expression for F(x) if F(x) = ( f ) (x). Be sure to list all g restrictions on the domain of F(x). f(x) = x 4 3x 54; g(x) = x 9 Answer: x + 6, x ±3 Sec 6.7: Synthetic Division A streamlined process of long division (synthetic division) can be used when the divisor has the form x c. Ex 8 Use synthetic division to divide. a) (x 3 4x + 5x 6) (x 3) b) (x 3 3x + 8) (x + ) RT says: RT says: Page 5 of 4

6 c) (8x 3 + 7x 6x ) (x ) d) PP (x5 43) (x 3) Answer: x 4 + 3x 3 + 9x + 7x + 8 RT says: The Remainder Theorem The remainder obtained upon dividing a polynomial P(x) by x c is P(c). What does the Remainder Theorem tell us about the above examples? Ex 9 Use synthetic division to find f( ) where f(x) = 3x 4 + 8x 3 + x 7x 4. Final? Page 6 of 4

7 Show a check: Ex 0 By the Factor Theorem, p(c) = 0 (where p is a polynomial) if and only if (x c) is a factor of p(x). Use synthetic division to show that (x + ) is a factor of (x 3 x + 6). Refer to example 7d. Show work and write conclusion using both the Remainder and Factor Theorems. Sec 6.: Rational Expressions and Functions: Adding and Subtracting & Sec 6.4: Rational Equations READ AND STUDY: ADDING/SUBTRACTING *Do not factor/cancel until AFTER combining numerators* IF cancelling occurs (which means the final numerator factored), be sure to list the restrictions. SOLVING CLEAR FRACTIONS and LIST any bad values (restrictions) and CHECK for vacuous answers. Warning: On exams, when solving rational equations, if we fail to recognize to clear fractions, NO partial credit for incorrect answers. Ex Simplify OR solve. a) DO b) DO 9 y 5 y 8 5ab a + b a + b a b Page 7 of 4

8 c) DO d) PP x + x + x 3 x + x 3 x 3 x + 4x + 3 x + 3x 0 Answer: 5x +3x+8 (x+5)(x )(x 4) 5x x 6x + 8 e) f) DO 5x x 6x + 8 3x + x x 3 x x + 30 x 6 = 9x 45 g) h) DO t 3 t 4 t = t 3 x 7 x 6 x 6 x Page 8 of 4

9 i) PP j) Start x 3 x = x 8 x + x 6 x x + 3 Ans: x = /3 t + 6 t = 5 Ans: t =, 3 k) DO l) x x + = 3 x x + x 6x + 8 x x x m) DO n) DO x x x 6 + x + 3 x 6 x x = 48 x 8x Page 9 of 4

10 o) DO x 3 x = x 9 x + x 6 x + x + 3 Ex a) Find all values for a such that f(a) = g(a). b) Find H(x) = f(x) g(x). f(x) = x 4 x 3 ; g(x) = x 3 x + x x + f(x) = x + 4 x + 5x + 6 ; g(x) = x + Ex 3 Find the simplified form for f(x) and list all restrictions on the domain. 3x f(x) = x + 5x 4 x 3 x 9 Page 0 of 4

11 Ex 4 PP (#6) Let f(x) = x+5 f(a) = g(a). Answer: 7 3 x +4x+3 and g(x) = x+ x 9 + x x x 3. Find all values of a for which PP Solve. x 4 x + 4 x 7x 4 = x + 4 x Start Which Fractions Reduce handout. Finish-IC or hw assignment. Ex 5 (# ) -6, -5 and 5, 6 Sec 6.5: Solving Applications Using Rational Equations The reciprocal of the product of two consecutive integers is. Find the two integers. 30 Ex 6 (#6) A community water tank can be filled in 8 hours by the town office well alone and in hours by the high school well alone. How long will it take to fill the tank if both wells are working? hrs Page of 4

12 Ex 7 (# 0) Kent can cut and split a cord of wood twice as fast as Brent can. When they work together, it takes them 4 hours. How long would it take each of them to do the job alone? B: hrs K: 6 hrs Ex 8 (# 36) The A train goes mph slower than the E train. The A train travels 30 miles in the same amount of time that the E train travels 90 miles. Find the speed of each train. A: 46mph; E: 58mph Ex 9 (# 44) Fiona s Boston Whaler cruised 45 miles upstream and 45 miles back in a total of 8 hours. The speed of the river is 3 mph. Find the speed of the boat in still water. mph Page of 4

13 Ex 0 Sec 6.8: Formulas, Applications, and Variation Solve for each specified variable. a) (# 0) b) (# 4) K = rt r t ; t p + q = f ; p Direct Variation The situation is modeled by a linear function of the form f(x) = kx, or y = kx, where k is a nonzero constant. We say there is direct variation, that y varies directly as x, or that y is proportional to x. Inverse Variation The situation is modeled by a rational function of the form f(x) = k/x, or y = k/x, where k is a nonzero constant. We say the there is inverse variation, that y varies inversely as x, or that y is inversely proportional to x. Joint Variation When a variable varies directly with more than one variable, we say that there is a joint variation. y varies jointly as x and z if for some nonzero constant k, y = kxz Note: When a variable varies directly and/or inversely at the same time with more than one other variable, there is a combined variation. Joint variation is a form of combined variation. The number k is called the variation constant, or constant of proportionality. Ex (# 46) Find the variation constant and an equation of variation if y varies directly as x and y = when x = 5. Ex (# 5) Find the variation constant and an equation of variation in which y varies inversely as x and y = 9 when x = 0. Page 3 of 4

14 Ex 3 (# 76) Find an equation of variation in which y varies directly as x and inversely as w and the square of z, and y = 4.5 when x = 5, w = 5, and z =. Ex 4 (# 80) The intensity I of a television signal varies inversely as the square of the distance from the transmitter. If the intensity is 5 W/m at a distance of km, what is the intensity 6.5 km from the transmitter? Grade each problem. Identify any and all mistakes. Problem: Simplify or solve. x + 5x + 6 x + 4 x + Solution : (x + )(x + 3) (x + )(x + 3) x + 4 (x + )(x + 3) x + Solution : (x + 4)(x + 3) LCD x + 7x + x + 7x + 4 Solution 3: (x + )(x + 3) x + 4 x + x + 7x + x + 7x + 0 (x + 5)(x + ) (x + )(x + 3) x + 5 x + 3 (x + 7x + ) LCD x 7x x 7x 0 LCD x 5 x + 3 = (x 5)(x + ) (x + )(x + 3) Page 4 of 4