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 + 3 40y 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
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 3 3 5 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 3 0 5 + 3 Method ONE 3 0 5 + 3 Page of 4
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 5 + 6 x + 5 e) f) In-Class Prob: Ans: y 7 x + 3 3 x + 6 7 x y y 3y y + 5y + 4 3y y y y + 3y 4 y+ Page 3 of 4
g) DO h) PP x x 7x + + 8 x 4 x + 4 0 5x 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
d) e) x 3 x + 6 x 3 8 x + x *answer only f) Practice Problem x 4 + 4x + 6 x + Answer: x + 3 + 3 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
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
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
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
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 8 + 6 x = 48 x 8x Page 9 of 4
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
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? 9 9 0 hrs Page of 4
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
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
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