# Polynomial and Rational Functions

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2 4 Chapter Polnomial and Rational Functions II. The Standard Form of a Quadratic Function (Pages 95 96) The standard form of a quadratic function is f() = a( h) + k, a 0. How to write quadratic functions in standard form and use the results to sketch graphs of functions For a quadratic function in standard form, the ais of the associated parabola is = h and the verte is (h, k). To write a quadratic function in standard form,... process of completing the square on the variable. use the To find the -intercepts of the graph of solve the equation a + b + c = 0. f ( ) = a + b + c,... Eample 1: Sketch the graph of f ( ) = + 8 and identif the verte, ais, and -intercepts of the parabola. ( 1, 9); = 1; ( 4, 0) and (, 0) III. Finding Minimum and Maimum Values (Pages 97 98) For a quadratic function in the form f ( ) = a + b + c, when a > 0, f has a minimum that occurs at b/(a). When a < 0, f has a maimum that occurs at b/(a). To find the minimum or maimum value, evaluate the function at b/(a). How to find minimum and maimum values of quadratic functions in real-life applications Eample : Homework Assignment Page(s) Eercises Find the minimum value of the function f ( ) = At what value of does this minimum occur? Minimum function value is 71/1 when = 11/6 Copright Houghton Mifflin Compan. All rights reserved.

3 Section. Polnomial Functions of Higher Degree 5 Section. Polnomial Functions of Higher Degree Objective: In this lesson ou learned how to sketch and analze graphs of polnomial functions. Course Number Instructor Date Important Vocabular Define each term or concept. Continuous The graph of a polnomial function has no breaks, holes, or gaps. Etrema The minimums and maimums of a function. Relative minimum The least value of a function on an interval. Relative maimum The greatest value of a function on an interval. Repeated zero If ( a) k, k > 1, is a factor of a polnomial, then = a is a repeated zero. Multiplicit The number of times a zero is repeated. Intermediate Value Theorem Let a and b be real numbers such that a < b. If f is a polnomial function such that f(a) f(b), then, in the interval [a, b], f takes on ever value between f(a) and f(b). I. Graphs of Polnomial Functions (Pages ) Name two basic features of the graphs of polnomial functions. 1) continuous ) smooth rounded turns How to use transformations to sketch graphs of polnomial functions Will the graph of 7 g ( ) = look more like the graph of f ( ) = or the graph of f ( ) =? Eplain. The graph will look more like that of f() = 3 because the degree of both is odd. II. The Leading Coefficient Test (Pages ) State the Leading Coefficient Test. As moves without bound to the left or to the right, the graph of the polnomial function f() = a n n a 1 + a 0 eventuall rises or falls in the following manner: 1. When n is odd: 3 How to use the Leading Coefficient Test to determine the end behavior of graphs of polnomial functions a. If the leading coefficient is positive, the graph falls to the left and rises to the right. b. If the leading coefficient is negative, the graph rises to the left and falls to the right.. When n is even: a. If the leading coefficient is positive, the graph rises to the left and right. b. If the leading coefficient is negative, the graph falls to the left and right. Copright Houghton Mifflin Compan. All rights reserved.

4 6 Chapter Polnomial and Rational Functions Eample 1: Describe the left and right behavior of the graph of 6 f ( ) = Because the degree is even and the leading coefficient is negative, the graph falls to the left and right. III. Zeros of Polnomial Functions (Pages ) Let f be a polnomial function of degree n. The function f has at most n real zeros. The graph of f has at most n 1 relative etrema. How to find and use zeros of polnomial functions as sketching aids Let f be a polnomial function and let a be a real number. List four equivalent statements about the real zeros of f. 1) = a is a zero of the function f ) = a is a solution of the polnomial equation f() = 0 3) ( a) is a factor of the polnomial f() 4) (a, 0) is an -intercept of the graph of f If a polnomial function f has a repeated zero = 3 with multiplicit 4, the graph of f touches the -ais at = 3. If f has a repeated zero = 4 with multiplicit 3, the graph of f crosses the -ais at = Eample : Sketch the graph of f ( ) = 3. IV. The Intermediate Value Theorem (Pages 111) Interpret the meaning of the Intermediate Value Theorem. If (a, f(a)) and (b, f(b)) are two points on the graph of a polnomial function f such that f(a) f(b), then for an number d between f(a) and f(b), there must be a number c between a and b such that f(c) = d. How to use the Intermediate Value Theorem to help locate zeros of polnomial functions Describe how the Intermediate Value Theorem can help in locating the real zeros of a polnomial function f. If ou can find a value = a at which f is positive and another value = b at which f is negative, ou can conclude that f has at least one real zero between a and b. Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.

5 Section.3 Real Zeros of Polnomial Functions 7 Section.3 Real Zeros of Polnomial Functions Objective: In this lesson ou learned how to use long division and snthetic division to divide polnomials b other polnomials and how to find the rational and real zeros of polnomial functions. Course Number Instructor Date Important Vocabular Define each term or concept. Long division of polnomials A procedure for dividing two polnomials, which is similar to long division in arithmetic. Division Algorithm If f() and d() are polnomials such that d() 0, and the degree of d() is less than or equal to the degree of f(), there eist unique polnomials q() and r() such that f() = d()q() + r() where r() = 0 or the degree of r() is less than the degree of d(). Snthetic division A shortcut for long division of polnomials when dividing b divisors of the form k. Remainder Theorem If a polnomial f() is divided b k, then the remainder is r = f(k). Factor Theorem A polnomial f() has a factor ( k) if and onl if f(k) = 0. Upper bound A real number b is an upper bound for the real zeros of f if no real zeros of f are greater than b. Lower bound A real number b is a lower bound for the real zeros of f if no real zeros of f are less than b. I. Long Division of Polnomials (Pages ) When dividing a polnomial f() b another polnomial d(), if the remainder r() = 0, d() divides evenl into f(). How to use long division to divide polnomials b other polnomials The rational epression f()/d() is improper if... the degree of f() is greater than or equal to the degree of d(). The rational epression r()/d() is proper if... of r() is less than the degree of d(). the degree Before appling the Division Algorithm, ou should... write the dividend and divisor in descending powers of the variable and insert placeholders with zero coefficients for missing powers of the variable. Eample 1: Divide b (13 + 4)/( + + 1) Copright Houghton Mifflin Compan. All rights reserved.

6 8 Chapter Polnomial and Rational Functions II. Snthetic Division (Page 119) Can snthetic division be used to divide a polnomial b 5? Eplain. No, the divisor must be in the form k. How to use snthetic division to divide polnomials b binomials of the form ( k) Can snthetic division be used to divide a polnomial b + 4? Eplain. Yes, rewrite + 4 as ( 4). Eample : Fill in the following snthetic division arra to 4 divide b 5. Then carr out the snthetic division and indicate which entr represents the remainder remainder III. The Remainder and Factor Theorems (Pages 10 11) To use the Remainder Theorem to evaluate a polnomial function f() at = k,... use snthetic division to divide f() b k. The remainder will be f(k). How to use the Remainder and Factor Theorems Eample 3: Use the Remainder Theorem to evaluate the 4 function f ( ) = at = To use the Factor Theorem to show that ( k) is a factor of a polnomial function f(),... use snthetic division on f() with the factor ( k). If the remainder is 0, then ( k) is a factor. Or, alternativel, evaluate f() at = k. If the result is 0, then ( k) is a factor. Copright Houghton Mifflin Compan. All rights reserved.

7 Section.3 Real Zeros of Polnomial Functions 9 List three facts about the remainder r, obtained in the snthetic division of f() b k: 1) The remainder r gives the value of f at = k. That is, r = f(k). ) If r = 0, ( k) is a factor of f(). 3) If r = 0, (k, 0) is an -intercept of the graph of f. IV. The Rational Zero Test (Pages 1 14) Describe the purpose of the Rational Zero Test. The Rational Zero Test relates the possible rational zeros of a polnomial with integer coefficients to the leading coefficient and to the constant term of the polnomial. How to use the Rational Zero Test to determine possible rational zeros of polnomial functions State the Rational Zero Test. If the polnomial f() = a n n + a n 1 n a + a 1 + a 0 has integer coefficients, ever rational zero of f has the form: rational zero = p/q, where p and q have no common factors other than 1, p is a factor of the constant term a 0, and q is a factor of the leading coefficient a n. To use the Rational Zero Test,... first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Then use trial and error to determine which of these possible rational zeros, if an, are actual zeros of the polnomial. Eample 4: List the possible rational zeros of the polnomial 5 function f ( ) = ± 1, ± 5, ± 1/3, ± 5/3 Some strategies that can be used to shorten the search for actual zeros among a list of possible rational zeros include... using a programmable calculator to speed up the calculations, using a graphing utilit to estimate the locations of zeros, using the Intermediate Value Theorem (along with a table generated b a graphing utilit) to give approimations of zeros, or using the Factor Theorem and snthetic division to test possible rational zeros, etc. 4 3 Copright Houghton Mifflin Compan. All rights reserved.

8 30 Chapter Polnomial and Rational Functions V. Other Tests for Zeros of Polnomials (Pages 14 16) State the Upper and Lower Bound Rules. Let f() be a polnomial with real coefficients and a positive leading coefficient. Suppose f() is divided b c, using snthetic division. 1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f.. If c < 0 and the numbers in the last row are alternatel positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f. How to use Descartes s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polnomials Eplain how the Upper and Lower Bound Rules can be useful in the search for the real zeros of a polnomial function. Eplanations will var. For instance, suppose ou are checking a list of possible rational zeros. When checking the possible rational zero with snthetic division, each number in the last row is positive or zero. Then ou need not check an of the other possible rational zeros that are greater than and can concentrate on checking onl values less than. Additional notes Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.

9 Section.4 Comple Numbers 31 Section.4 Comple Numbers Objective: In this lesson ou learned how to perform operations with comple numbers and plot comple numbers in the comple plane. Course Number Instructor Date Important Vocabular Define each term or concept. Comple number If a and b are real numbers, the number a + bi, where the number a is called the real part and the number bi is called the imaginar part, is a comple number written in standard form. Comple conjugates A pair of comple numbers of the form a + bi and a bi. Imaginar ais The vertical ais in the comple plane. Real ais The horizontal ais in the comple plane. Bounded A description for a sequence in which the absolute value of each number in the sequence is less than some fied number N. Unbounded A description for a sequence in which the absolute values of the terms of the sequence become infinitel large. I. The Imaginar Unit i (Page 131) Mathematicians created an epanded sstem of numbers using the imaginar unit i, defined as i = 1, because... there is no real number that can be squared to produce 1. How to use the imaginar unit i to write comple numbers B definition, i = 1. For the comple number a + bi, if b = 0, the number a + bi = a is a(n) real number. If b 0, the number a + bi is a(n) imaginar number. If a = 0, the number a + bi = b, where b 0, is called a(n) pure imaginar number. The set of comple numbers consists of the set of real numbers and the set of imaginar numbers. Two comple numbers a + bi and c + di, written in standard form, are equal to each other if... and onl if a = c and b = d. Copright Houghton Mifflin Compan. All rights reserved.

10 3 Chapter Polnomial and Rational Functions II. Operations with Comple Numbers (Pages ) To add two comple numbers,... add the real parts and the imaginar parts of the numbers separatel. How to add, subtract, and multipl comple numbers To subtract two comple numbers,... subtract the real parts and the imaginar parts of the numbers separatel. The additive identit in the comple number sstem is 0. The additive inverse of the comple number a + bi is (a + bi) = a bi. Eample 1: Perform the operations: (5 6i) (3 i) + 4i To multipl two comple numbers a + bi and c + di,... the multiplication rule (ac bd) + (ad + bc)i or use the use Distributive Propert to multipl the two comple numbers, similar to using the FOIL method for multipling two binomials. Eample : Multipl: (5 6i)(3 i) 3 8i III. Comple Conjugates (Page 134) The product of a pair of comple conjugates is a(n) real number. To find the quotient of the comple numbers a + bi and c + di, where c and d are not both zero,... multipl the numerator and denominator b the comple conjugate of the denominator. How to use comple conjugates to write the quotient of two comple numbers in standard form Eample 3: Divide (1 + i) b ( i). Write the result in standard form. 1/5 + 3/5i Copright Houghton Mifflin Compan. All rights reserved.

11 Section.4 Comple Numbers 33 IV. Fractals and the Mandelbrot Set (Pages ) The comple plane is... a coordinate sstem in which ever point corresponds to a comple number a + bi. How to plot comple numbers in the comple plane On the comple plane shown below, (a) label the real ais, (b) label the imaginar ais, and (c) plot and label the comple numbers 3i and 4 + i. Imaginar ais i Real ais 3i -3-5 Let c represent a comple number. Describe how to tell whether or not c belongs to the Mandelbrot Set. Use the value of c to find the sequence c, c + c, (c + c) + c, [(c + c) + c] + c,. If the sequence is bounded, the comple number c is in the Mandelbrot Set. If the sequence is unbounded, the comple number c is not in the Mandelbrot Set. Describe how the Mandelbrot Set could be graphed. Descriptions ma var. For instance, one wa to graph the Mandelbrot Set would be to plot points in the comple plane with one of two colors: one color for points that are in the set and another color for points that are not in the set. Copright Houghton Mifflin Compan. All rights reserved.

13 Section.5 The Fundamental Theorem of Algebra 35 Section.5 The Fundamental Theorem of Algebra Objective: In this lesson ou learned how to determine the numbers of zeros of polnomial functions and find them. Course Number Instructor Date Important Vocabular Define each term or concept. Fundamental Theorem of Algebra If f() is a polnomial of degree n, where n > 0, then f has at least one zero in the comple number sstem. Linear Factorization Theorem If f() is a polnomial of degree n, where n > 0, f has precisel n linear factors f() = a n ( c 1 )( c )... ( c n ) where c 1, c,..., c n are comple numbers. Conjugates A pair of comple numbers of the form a + bi and a bi. I. The Fundamental Theorem of Algebra (Pages ) In the comple number sstem, ever nth-degree polnomial function has precisel n zeros. Eample 1: How man zeros does the polnomial function f ( ) = have? How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polnomial function and find all zeros of polnomial functions, including comple zeros An nth-degree polnomial can be factored into precisel n linear factors. Eample : List all of the zeros of the polnomial function 3 f ( ) = , 6i, 6i II. Conjugate Pairs (Page 141) Let f() be a polnomial function that has real coefficients. If a + bi, where b 0, is a zero of the function, then we know that a bi is also a zero of the function. How to find conjugate pairs of comple zeros Copright Houghton Mifflin Compan. All rights reserved.

14 36 Chapter Polnomial and Rational Functions III. Factoring a Polnomial (Pages ) To write a polnomial of degree n > 0 with real coefficients as a product without comple factors, write the polnomial as... the product of linear and/or quadratic factors with real coefficients, where the quadratic factors have no real zeros. How to find zeros of polnomials b factoring A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Eample 3: Write the polnomial f ( ) = (a) as the product of linear factors and quadratic factors that are irreducible over the reals, and (b) in completel factored form. (a) f() = ( + )( )( + 9) (b) f() = ( + )( )( + 3i)( 3i) 4 Eplain wh a graph cannot be used to locate comple zeros. Real zeros are the onl zeros that appear as -intercepts on a graph. A polnomial function s comple zeros must be found algebraicall. Additional notes Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.

15 Section.6 Rational Functions and Asmptotes 37 Section.6 Rational Functions and Asmptotes Objective: In this lesson ou learned how to determine the domains and find asmptotes of rational functions. Course Number Instructor Date Important Vocabular Define each term or concept. Rational function A function that can be written in the form: f() = N()/D(), where N() and D() are polnomials and D() is not the zero polnomial. Vertical asmptote The line = a is a vertical asmptote of the graph of f if f() or f() as a, either from the right or from the left. Horizontal asmptote The line = b is a horizontal asmptote of the graph of f if f() b as or. I. Introduction to Rational Functions (Pages ) The domain of a rational function of includes all real numbers ecept... -values that make the denominator zero. How to find the domains of rational functions To find the domain of a rational function of,... set the denominator of the rational function equal to zero and solve for. These values of must be ecluded from the domain of the function. 1 Eample 1: Find the domain of the function f ( ) =. 9 The domain of f is all real numbers ecept = 3 and = 3. II. Horizontal and Vertical Asmptotes (Pages ) The notation f() 5 as means... that f() approaches 5 as increases without bound. How to find horizontal and vertical asmptotes of graphs of rational functions Let f be the rational function given b N( ) an f ( ) = = D( ) b m n m + a + b n 1 m 1 n 1 m a + a b + b where N() and D() have no common factors. 1) The graph of f has vertical asmptotes at the zeros of D(). 0 Copright Houghton Mifflin Compan. All rights reserved.

16 38 Chapter Polnomial and Rational Functions ) The graph of f has at most one horizontal asmptote determined b comparing the degrees of N() and D(). a) If n < m, the line = 0 (the -ais) is a horizontal asmptote. b) If n = m, the line = a n /b m is a horizontal asmptote. c) If n > m, the graph of f has no horizontal asmptote. Eample : Find the asmptotes of the function 1 f ( ) =. 6 Vertical: =, = 3; Horizontal: = 0 III. Applications of Rational Functions (Pages ) Give an eample of asmptotic behavior that occurs in real life. Answers will var. How to use rational functions to model and solve real-life problems Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.

17 Section.7 Graphs of Rational Functions 39 Section.7 Graphs of Rational Functions Objective: In this lesson ou learned how to sketch graphs of rational functions. Course Number Instructor Date Important Vocabular Define each term or concept. Slant (or oblique) asmptote If the degree of the numerator of a rational function is eactl one more than the degree of the denominator, then the line determined b the quotient of the denominator into the numerator is a slant asmptote of the graph of the rational function. I. The Graph of a Rational Function (Pages ) To sketch the graph of the rational function f() = N()/D(), where N() and D() are polnomials with no common factors,... 1) Simplif f, if possible. An restrictions on the domain of f not in the simplified function should be listed. ) Find and plot the -intercept (if an) b evaluating f(0). 3) Find the zeros of the numerator (if an) b setting the numerator equal to zero. Then plot the corresponding - intercepts. 4) Find the zeros of the denominator (if an) b setting the denominator equal to zero. Then sketch the corresponding vertical asmptotes using dashed vertical lines and plot the corresponding holes using open circles. 5) Find and sketch an other asmptotes of the graph using dashed lines. 6) Plot at least one point between and one point beond each -intercept and vertical asmptote. 7) Use smooth curves to complete the graph between and beond the vertical asmptotes, ecluding an points where f is not defined. How to analze and sketch graphs of rational functions Eample 1: Sketch the graph of 3 f ( ) =. + 4 Copright Houghton Mifflin Compan. All rights reserved.

18 40 Chapter Polnomial and Rational Functions II. Slant Asmptotes (Page 159) To find the equation of a slant asmptote,... use long division to divide the denominator of the rational function into the numerator. The equation of the slant asmptote is the quotient, ecluding the remainder. How to sketch graphs of rational functions that have slant asmptotes Eample : Decide whether each of the following rational functions has a slant asmptote. If so, find the equation of the slant asmptote (a) f ( ) = (b) f ( ) = (a) Yes, = 3 (b) No III. Applications of Graphs of Rational Functions (Page 160) Describe a real-life situation in which a graph of a rational function would be helpful when solving a problem. How to use rational functions to model and solve real-life problems Answers will var. Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.

20 4 Chapter Polnomial and Rational Functions III. Choosing a Model (Pages ) If it isn t eas to tell from a scatter plot which tpe of model a set of data would best be modeled b, ou should... first find several models for the data and then choose the model that best fits the data b comparing the -values of each model with the actual -values. How to choose a model that best fits a set of data Homework Assignment Page(s) Eercises Copright Houghton Mifflin Compan. All rights reserved.

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