# Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Save this PDF as:

Size: px
Start display at page:

Download "Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions."

## Transcription

1 Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and b be real numbers, and let m and n be integers. Product of Powers Propert Quotient of Powers Propert m n m n a a a m m a mn a 1 a or, a 0 n n nm a a a m Power of a Power Propert n a a mn ab a b Power of a Product Propert m m m m a a Power of a Quotient Propert, b 0 m b b m Negative Eponent Propert Zero Eponent Propert n m 1 a b a m or, a 0 a b a a 0 1, a 0 n Evaluating Numerical Epressions with Eponents E 1: Evaluate. Use the power of a power propert: Use the negative eponent propert: E : Evaluate. Use the power of a product propert: Use the negative eponent propert: Use the power of a quotient propert: 9 4 Page 1 of McDougal Littell

2 E : Evaluate 6 0. Algebra II Notes Unit Si: Polnomials Use the zero eponent propert: Use the quotient of a power propert: 6 Use the negative eponent propert: Note: We can use the quotient of a power propert to keep the eponent positive. Use the quotient of a power propert: Simplifing Algebraic Epressions E 1: Simplif the epression 8 1 and write with positive eponents. Use the power of a product propert: 1 Use the power of a power propert: Use the product of a power propert: Use the negative eponent propert: E : Simplif the epression 0 t v and write with positive eponents. Use the zero eponent propert: t 1 t 1 Use the negative eponent propert: t t t E : Simplif the epression 1 6 Use the product of a power propert: Use the power of a power propert: Use the quotient of a power propert: Simplif: 4 and write with positive eponents Page of McDougal Littell

3 Algebra II Notes Unit Si: Polnomials You Tr: Simplif the epression and write with positive eponents. Identif which properties of eponents ou used QOD: Which properties of eponents require ou to check that two or more bases are the same before appling the propert? Sample SAT Question(s): Taken from College Board online practice problems If m is a positive integer, which of the following is NOT equal to m? (A) (B) 4 m 4 m (C) m m (D) 4 m (E) Grid-In 16 m. For all positive integers a and b, let a b be defined b b a 1 ab a 1. What is the value of 4?. The positive integer n is not divisible b 7. The remainder when n is divided b are each equal to k. What is k? (A) 1 (B) (C) 4 (D) 6 (E) It cannot be determined from the information given. Page of McDougal Littell n is divided b 7 and the remainder when

4 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6.1 The student will graph a polnomial function with and without technolog. f a a... a a a, a 0 n n1 Polnomial Function: a function of the form n n1 1 0 Note: Eponents are whole numbers and coefficients are real numbers. n Leading Coefficient: a n Constant Term: a 0 Degree: n Note: Consider using the Fraer Model (Vocabular Concept Grid) Activit See resource pages. Standard Form of a Polnomial Function: The terms are written in descending order of the eponents Names of Polnomial Functions: This is kind of trick but a n is the name of the coefficient with the same degree. So, a n is the coefficient of the term that is the n th degree and a n 1 is the coefficient of the term that is degree n 1. Degree Tpe Standard Form 0 Constant f a0 1 Linear f a1 a0 Quadratic f a a a 1 0 Cubic f a a a a Quartic 4 f a a a a a Identifing Polnomial Functions f 8 a polnomial function? If es, write it in standard form. E 1: Is No. In order to be a polnomial function, all eponents must be whole numbers. f 8 a polnomial function? If es, write it in standard form. E : Is 4 Yes. All eponents are whole numbers and all coefficients are real numbers. f 8 Note: This is a quartic trinomial (degree = 4). Standard Form: 4 Evaluating Polnomial Functions Using Direct Substitution E 1: Find 4 f if f f So : f () 47 Page 4 of McDougal Littell

5 Algebra II Notes Unit Si: Polnomials Evaluating Polnomial Functions Using Snthetic Substitution E 1: Find 4 f if f 6 1 using snthetic substitution. Using the polnomial in standard form, write the coefficients in a row. Put the -value to the upper left Bring down the first coefficient, then multipl b the -value. multipl Add straight down the columns, and repeat The number in the bottom right is the value of So : f () 47 f. E : Find f if f 7 11 using snthetic substitution. This polnomial function is in standard form, however it is missing two terms. We can rewrite the f to fill in the missing terms. function as This also means that (, 17) is an ordered pair that would be a point on the graph. f 17 Graphing Polnomial Functions: To graph a polnomial function, make a table of values using snthetic substitution, plot the points, and determine the end behavior to draw the rest of the graph. End Behavior: the behavior of the graph as gets ver large (approaches positive infinit ) OR as gets ver small (or approaches negative infinit ). Notation: ( approaches positive infinit ) (The ver far right end of a graph). ( approaches negative infinit) (The ver far left end of a graph). Page of McDougal Littell

6 Algebra II Notes Unit Si: Polnomials Eploration Activit: Graph each function on the calculator. Determine the end behavior of f as approaches negative and positive infinit. Fill in the table and write our conclusion regarding the degree of the function and the end behavior. (Teacher Note: Answers are in red.) f Degree Sign of Leading Coefficient f f + + f + f + + f + 4 f f 4 + f + + f + 6 f f Page 6 of McDougal Littell

7 Conclusion: The graph of a polnomial function n n 1 f an an 1... a a1 a0 has the following end behavior. These patterns are ver predicatable. Algebra II Notes Unit Si: Polnomials Degree Lead Coefficient End Behavior Even Positive as, f as, f Even Negative as, f as, f Think of end behavior as what happens on either end of the graph. There can be a lot of curves, etc. in the middle, but polnomial functions either increase or decrease at the far ends (as, f( ) ). Odd Positive as, f as, f Odd Negative as, f as, f E 1: Graph the polnomial function 4 f 1 b hand. Check our graph on the graphing calculator. Step One: Make a table of values using snthetic substitution f Step Two: Determine end behavior using the degree and sign of the leading coefficient. The degree is even, and the leading coefficient is positive. So as, f as, f. Step Three: Graph the polnomial function Page 7 of McDougal Littell

8 Algebra II Notes Unit Si: Polnomials E : Graph the polnomial function f 4 b hand. Check our graph on the graphing calculator. Step One: Make a table of values using snthetic substitution f Wh didn t we use snthetic subsitution to find f(0)? Step Two: Determine end behavior using the degree and sign of the leading coefficient. The degree is odd and the leading coefficient is negative. So. Step Three: Graph the polnomial function. 1 as, f as, f You Tr: Graph the polnomial function f 1b hand. Check our graph on the graphing calculator. QOD: Which term of the polnomial function is most important when determining the end behavior of the function? Page 8 of McDougal Littell

9 Sample CCSD Common Eam Practice Question(s): Algebra II Notes Unit Si: Polnomials Which best represents the graph of the polnomial function? 4 Page 9 of McDougal Littell

10 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6. The student will simplif polnomial epressions. Adding Polnomials E 1: Add the polnomials Subtracting Polnomials. Vertical Method: Write each polnomial in standard form and line up like terms. Then add the like terms E 1: Subtract the polnomials To subtract, we will rewrite the problem as an addition problem b adding the opposite Horizontal Method: Combine each set of like terms Write the final answer in standard form Multipling Polnomials E 1: Find the product 4. Horizontal Method: Use the distributive propert b distributing each term of the first polnomial Combine like terms and write the answer in standard form Page 10 of McDougal Littell

11 Algebra II Notes Unit Si: Polnomials E : Multipl the polnomials Vertical Method: Use long multiplication E : Multipl the polnomials 14. Multipl the polnomials two at a time. Because the are binomials, we can use FOIL to multipl the first two Use the distributive propert Combine like terms and write in standard form Review: Special Products (Allow students to come up with these on their own.) Memorize these! Sum and Difference Product a bab a b Square of a Binomial Cube of a Binomial a b a abb a b a abb a b a a bab b a b a a bab b E 1: Simplif the epression. Using the cube of a binomial: Page 11 of McDougal Littell

12 Application Problems Algebra II Notes Unit Si: Polnomials E 1: Find a polnomial epression for the volume of a rectangular prism with sides, 4, and. Volume of a Rectangular Prism = Length Width Height FOIL: 4 1 Vertical Method: E : From 198 through 1996, the number of flu shots given in one cit can be modeled b A t t t t for adults and b C t t t t Write a model for the total number F of flu shots given in these ears for children, where t is the number of ears since 198. To find the total flu shots, we need to add the polnomials. Vertical Method: Solution: 4 11.t 8.t 194t 4190t t 106t 1t 1t t 97.67t 194t 40t 81 F t t t t You Tr: Find the product: 7 1 QOD: What is the advantage of the vertical method when adding, subtracting, or multipling polnomials? Sample CCSD Common Eam Practice Question(s):? Which polnomial represents the product of 8 A. B. C. D Page 1 of McDougal Littell

13 Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will solve polnomial equations b factoring and graphing. Review: Factoring Patterns Factoring a General Trinomial E 1: Factor the trinomial 1. ac Method: ac 4 Split the middle term: Factor b grouping: Factoring a Perfect Square Trinomial E 1: Factor the trinomial 6 9. Difference of Two Squares Use a abb a b. E 1: Factor Common Monomial Factor Use a b aba b E 1: Factor the trinomial completel. 7 Factor the GCF and the binomial square Since this is not completel factored, use a b a ba b Sum and Difference of Two Cubes 4 a b ab a abb a b ab a abb. Page 1 of McDougal Littell

14 E 1: Factor the binomial Algebra II Notes Unit Si: Polnomials 8 1. Use a b a ba ab b E : Factor the binomial 7. 6 Use a b a ba ab b. 4 9 Factoring b Grouping E 1: Factor the polnomial Group each pair of terms and factor the GCF. 9 Factor the common binomial factor. 9 Factor the remaining terms if possible. Review: Zero-Product Propert If ab 0, then a 0 or b 0. Solving Polnomial Equations b Factoring E 1: Solve the equation Step One: Set the equation equal to zero. Step Two: Factor the polnomial Step Three: Set each factor equal to zero and solve. Solutions:,,0,, , 0, Page 14 of McDougal Littell

15 Algebra II Notes Unit Si: Polnomials E : Find the real-number solutions of the equation 0. Step One: Set the equation equal to zero. Step Two: Factor the polnomial Step Three: Set each factor equal to zero and solve. Real-Number Solution: 0 0, i i Application Problem E 1: An optical compan is going to make a glass prism that has a volume of 1 cm. The height will be h cm, and the base will be a right triangle with legs of length h cm and h cm. What will be the height? 1 h h h Volume of a Prism = Area of the Base Height 1 To solve this equation for h, we must set it equal to zero. 1 h h h 1 h h h 1 h h h Before factoring, we can multipl both sides of the equation b to eliminate (clear) the fractions. 1 0 h h 6h 1 0 h h 1h 0 Factor b grouping. 0 h h 6 h h h 0 6 Solve b setting each factor equal to 0. The height of the prism will be cm. h 0 h cm h h 60 6 No real solution Page 1 of McDougal Littell

16 Algebra II Notes Unit Si: Polnomials You Tr: Solve the equation QOD: Give an eample of a binomial that can be factored either as the difference of two squares or as the difference of two cubes. Show the complete factorization of our binomial. Sample CCSD Common Eam Practice Question(s): 1. Which of the following represents the solution set of the polnomial equation A.,, i, i B.,,, C., i, i i, i D., i, i, 7 1 0? 4. What is the factored form of the polnomial A. 9 B. 9 C. 9 D. 9 7?. Which lists the set of all real zeros of the following polnomial function? A. B., f 4 1 C.,, D.,,1, Page 16 of McDougal Littell

17 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6.6 The student will divide polnomials and relate the result to the remainder theorem and the factor theorem. Dividing Polnomials Using Long Division On Your Own: Find the quotient of 1,6 and 4 using long division. On Your Own: Find the quotient of and 4 1. For each step of long division, we will divide the term with the highest power in the dividend b the first term of the divisor Remember to put a place for the missing term. 4 (add the opposite) (bring down the net term) (remainder) Eploration: Use the polnomial function f. Use long division to divide f. Then use snthetic substitution to find f. What do ou notice? b Remainder Theorem: If a polnomial is f divided b k, then the remainder is r f k. Dividing Polnomials Using Snthetic Division (Note: This can onl be used when the divisor is in the form k.) E 1: Divide the polnomial 7 6 b. Use snthetic substitution for k. The coefficients of the quotient and remainder appear in snthetic substitution. Quotient: R Note for graphing: This means that (, 7) is an ordered pair that is on the graph of the function. Page 17 of McDougal Littell

18 Factor Theorem: A polnomial Algebra II Notes Unit Si: Polnomials f has a factor k if and onl if f k 0. E 1: Factor f given that f 6 0. Because f 6 0, we know that 6 is a factor of f snthetic division to find the other factors E : One zero of f 6 f b the Factor Theorem. We will use f 9 1 is 7. Find the other zeros of the function. f 7 0, we know that 7 Because snthetic division to find the other factors. is a factor of f Note for graphing: This means that ( 6,0) is an ordered pair that is on the graph of the function. 6 is called a zero. It is also an intercept. b the Factor Theorem. We will use f 7 1 f Set each factor equal to zero and 4 You Tr: Use long division to find the quotient of You Tr: Given that t is a zero of the function. f t 4t 9t t 1, find the other zeros. QOD: If f is a polnomial that has a as a factor, what do ou know about the value of f a? Sample CCSD Common Eam Practice Question(s): What is 9 divided b? A. B. C. D Page 18 of McDougal Littell

19 Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6.7 The student will identif all possible rational zeros of a polnomial function b using the rational root theorem. 6.4 The student will find rational zeros of a polnomial. Using the Rational Zero Theorem Review: Rational zero is a rational number that produces a function value of 0. It can be visualized as f( ) 0 where is a rational number. On the graph it is an -intercept. The Rational Zero Theorem If f ( ) a n n... a1a0has integer coefficients, then ever rational zero of p factor of constant term a0 f has the following form: q factor of leading coefficient an The first important step is to list the possible rational zeros. After the are listed we can test them to determine if the are rational zeros. If the value of the possible rational zeros =0, the are called zeros. List the possible rational zeros: E 1: Find the possible rational zeros of f ( ) 7 1 Step 1: The leading coefficient is 1. 1 is the onl factor of 1. Step : The constant is 1. All of the factors of 1 are 1,,, 4, 6, 1. Step : List the possible factors ,,,,,and *If we tested for actual zeros using snthetic substitution from previous lessons we would find that and 4 are zeros. This also means that could not be a zero, 7 could not be a zero, 1 could not be a zero. E : Find the possible rational zeros of f ( ) 7 1 Step 1: The leading coefficient is. The factors of are 1 and. Step : The constant is -. All of the factors of - are, Step : List the possible factors -,,, 1 1 *We will not test for actual zeros for this eample. This also means that could not be a zero, 4 could not be a zero, could not be a zero. When the leading coefficient is not 1, the list of possible zeros can increase dramaticall. There are man tools that are used to find the rational zeros. We know some of those tools now and others will be introduced later in the class. Eamples that follow will demonstrate some of them. Page 19 of McDougal Littell

20 Algebra II Notes Unit Si: Polnomials f 7 18 E : Find all the real zeros of f 18 7 Step 1: Put the function in standard order. Step : List possible rational zeros (1,,,4,6,8,9,1,18,4,6,7) Step : Tr the possible zeros until ou find one From previous lessons, the function can be reduced to: Then factored: 4 f f 6 ( 4) Zeros: ( 4) Note: Finding rational zeros is also referred to as finding real zeros. Rational numbers are also real numbers. There is a distinction between listing possible rational zeros and finding rational (real) zeros. f E 4: Find all of the real zeros of Step 1: Notice that each term contains a common factor of. The problem can be factored to f ( 4 6) and since 0onl ( 4 6) can be =0. Step : List possible rational zeros (1,,,6) (Since the leading coefficient is now 1) Step : Tr the possible zeros until ou find one From previous lessons, the function can be reduced to: Then factored: f f ( 1) 0 0 ( 1) 0 Zeros: 1 Page 0 of McDougal Littell

21 E : Find all the real zeros of 4 Algebra II Notes Unit Si: Polnomials f 4 81 Step 1: Mabe this could be graphed first. Step : Look at the graph for reasonable choices 1 It appears the might be,, and Step : Check the chosen values using snthetic division. Start with -. Wh not ½?? Wh not 7? Is a root (zero). f ( )( ) The factored form so far is 4 Step 4: Repeat the steps above using a different reasonable choice. Tr Is a root (zero) Step : Repeat the steps above using a different reasonable choice. Tr Is a root (zero). Step 6: The function 7 is left to be factored. ( 9) has no real factors. 1 Solution: There are real zeros:,, and 1 Yes, all three work! And each time, the function (polnomial) is reduced b one degree. f You Tr: Find all real zeros of QOD: If the leading coefficient of a polnomial with integer coefficients is 1, what tpe of numbers must an possible rational zeros be? Page 1 of McDougal Littell

22 Sample CCSD Common Eam Practice Question(s): Algebra II Notes Unit Si: Polnomials Which lists the set of all real zeros of the following polnomial function? A. B., C.,, D.,,1, f 4 1 Page of McDougal Littell

23 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6. The student will use the Fundamental Theorem of Algebra to determine the number of zeros of a polnomial function. The Fundamental Theorem of Algebra If f is a polnomial of degree n where n0, then the equation f( ) 0 has at least one root in the set of comple numbers. Finding the number of solutions or zeros Review: Find the solutions of the following eamples. State how man solutions each has and classif each zero as rational, irrational, or comple (imaginar). E 1: 1 0 E : 9 0 E : 1 0 for the quadratic factor.) (Hint: Use the factorization for the difference of cubes, then use the quadratic formula Do ou notice a pattern with the degree of the polnomial and the number of solutions each has? E 4: How man different solutions are there to How do ou eplain this number? ? E : How man different solutions are there to Solution: 4, i 4i 16 0? Note: On the graph, the imaginar roots do not cross the ais. Note: 4, i 4i are comple conjugate pairs. 1i,1 i are comple conjugate pairs. The comple roots of polnomial functions with real coefficients alwas occur in comple conjugate pairs. Is this also the case for irrational zeros? Page of McDougal Littell

24 Algebra II Notes Unit Si: Polnomials Finding the zeros of a polnomial function This activit involves finding the rational zeros as learned in the previous section, then using other tools, such as the quadratic formula or technolog, to find the irrational or comple roots. E 1: Find all zeros of f 4 ( ) Using the rational root theorem and snthetic division, it can be shown that is a repeated root and and -1 are roots. The factored form looks like this: ( ) ( )( 1). The graph is shown. When a factor k is raised to an odd power, the graph crosses through the -ais. When a factor k is raised to an even power, the graph is tangent to the -ais. Solution: There are four real zeros is a repeated root and and -1 are roots. E : Find all zeros of f 4 ( ) 0 Using the rational root theorem and snthetic division, it can be shown that and - are roots. 4 Using the pattern of E it can be shown that f( ) 0 factors to ( )( )( ) Using the quadratic formula ields zeros of = i Solution: There are four zeros, and - and i. Two are real and two are comple conjugates Using Zeros to Write Polnomial Functions E 1: Write a polnomial function f of least degree that has real coefficients, a leading coefficient of 1, and, and - as zeros. Step 1: Write f ( ) in factored form: f( ) ( )( )( ) Step : Review - Multipl the polnomials two at a time. Because the are binomials, we can use FOIL to multipl the first two. f( ) ( 6)( ) f( ) 190 Solution: f ( ) 19 0 Page 4 of McDougal Littell

25 Algebra II Notes Unit Si: Polnomials E : Write a polnomial function f of least degree that has real coefficients, a leading coefficient of 1, and 1, -, and 1- i as zeros. Step 1: Since 1- i is a zero, so is 1+ i Step : Write f ( ) in factored form: f ( ) ( 1)( )( (1- i) ( (1+ i)) Step : Regroup: f ( ) ( 1)( ) ( 1)- i ( 1)+ i 10 Step 4: Epand, multipl polnomials, and combine like terms. f( ) ( ) ( 1) - i f ( ) ( ) ( 1 1 f ( ) ( )( ) f 4 ( ) Note: This graph onl has two intercepts. Wh? Using Technolog to Approimate Zeros Specific instructions should be given based on the calculator used. This section will provide onl general direction. E 1: 4 Approimate the real zeros of f( ) Use a graphing calculator to graph and calculate the zeros. You Tr: State the number of zeros of f 1 and find what the are. QOD: What is the conjugate of a comple number, and wh is it important when finding all of the zeros of a polnomial function? Page of McDougal Littell

26 Sample CCSD Common Eam Practice Question(s): Algebra II Notes Unit Si: Polnomials According to the Fundamental Theorem of Algebra, how man comple zeros does the polnomial f 1 have? A. B. C. 4 D. 4 Page 6 of McDougal Littell

27 Algebra II Notes Unit Si: Polnomials Sllabus Objective: 6.8 The student will analze graphs of polnomial functions to determine its characteristics. Analzing polnomial graphs Concept Summar n n 1 Let f an an 1... a a1a0 be a polnomial function. The following statements are equivalent: Zero: k is a zero of the function f. Factor: Solution: k is a factor of polnomial f(). k is a solution of the polnomial function f()=0. - Intercept: k is an -intercept of the graph of the polnomial function f. Using -Intercepts to Graph a Polnomial Function f 1 E 1: Graph the function Step 1: Plot the -intercepts. Since + and 1 are factors, and 1 are zeros (-intercepts) Note: + is raised to an odd power so the graph crosses the -ais at =. 1 is raised to an even power so the graph is tangent to the -ais at = 1. When a factor k is raised to an odd power, the graph crosses through the -ais. When a factor k is raised to an even power, the graph is tangent to the -ais. Step : Plot a few points between the -intercepts. f(0) ; f( 1) 4 Step : Determine the end behavior of the graph. Cubic function (odd degree) with positive leading coefficient as, f as, f Step 4: Sketch the graph Page 7 of McDougal Littell

28 Finding Turning Points Algebra II Notes Unit Si: Polnomials Turning points of polnomial functions: Another important characteristic of graphs of polnomial functions is that the have turning points corresponding to local maimum and minimum values. The coordinate of a turning point is a local maimum if the point is higher than all nearb points. The coordinate of a turning point is a local minimum if the point is lower than all nearb points. The graph of ever polnomial function of degree n has at most n 1 turning points. Moreover, if a polnomial has n distinct real zeros, then its graph has eactl n 1 turning points. E 1: Identif the zeros and turning points (estimate the zeros and turning points) Turning point (ma) zero zero Turning point (min) Turning point (ma) zero Turning point (min) Leading coefficient positive real zeros (including the double zero) {, 1, 1} turning points ( 1,4); (1,0) 1 local ma; 1 local min Leading coefficient positive 1 real zero, imaginar zeros {} turning points (0, ); (, 8) 1 local ma; 1 local min You Tr: Leading coefficient real zeros -10 Leading coefficient real zeros -10 Leading coefficient real zeros -10 Leading coefficient real zeros turning points turning points turning points turning points local ma; local min local ma; local min local ma; local min local ma; local min Page 8 of McDougal Littell

29 Algebra II Notes Unit Si: Polnomials E : Use a graphing calculator to graph and calculate the approimate local maimum(s) and local minimum(s) of f( ) ( )( )( ) Local maimum Coordinates:( 1, 6) Local ma is 6 at = 1 Local minimum You Tr: Graph the function use the calculator to find the turning points of the function. You Tr: Graph the function f 1 b hand. Check our graph on the graphing calculator and f 9 1 on the calculator and find the local etrema. QOD: What is the difference between local and absolute maima and minima? Sample CCSD Common Eam Practice Question(s): 1. Which describes the end behavior of the graph of 4 A. f B. f C. f 0 D. f f as? Page 9 of McDougal Littell

30 Algebra II Notes Unit Si: Polnomials. Use the graph of a polnomial function below. A. {} What are the zeros of the polnomial? B. { } C. {, 1, 4} D. {, 1, 4} Eploring Data and Statistics (Notes are not provided for this material) Modeling with Polnomial Functions Write a polnomial function whose intercepts are given. Finding and Using Finite Differences Properties of finite differences 1. If a polnomial function f() has degree n, then the n th -order differences of function values for equall spaced values are non-zero and constant.. Conversel, if the n th order differences of equall spaced data are non-zero and constant, then the data can be represented b a polnomial function of degree n. Polnomial Modeling with Technolog Graphing calculators make it eas to enter data, make a scatter plot, and calculate linear, quadratic, cubic, and quartic regressions. Page 0 of McDougal Littell

31 Unit Summar: Algebra II Notes Unit Si: Polnomials Polnomial equations provide some of the most classic problems in all of algebra. Finding zeros and etrema have man real-world applications. Real-life situations are modeled b writing equations based on data and using those equations to determine or estimate other data points (speed, volume, time, profits, patterns, etc). Page 1 of McDougal Littell

### 2.1 Evaluate and Graph Polynomial

2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of

### Algebra I Notes Unit Eleven: Polynomials

Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials

### SEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p.

Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of

### Factoring Polynomials

5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.D 2A.7.E Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its

### Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

### Algebra 1 Skills Needed to be Successful in Algebra 2

Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed

### Unit 2 Notes Packet on Quadratic Functions and Factoring

Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

### Algebra/Pre-calc Review

Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge

### Summer Review For Students Entering Algebra 2

Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.

Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions - Polnomial Functions and Their Graphs Polnomial Functions: - a function that consists of a polnomial epression

### + = + + = x = + = + = 36x

Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

### Graphing Calculator Computations 2

Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)

### INTRODUCTION GOOD LUCK!

INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills

### Polynomial and Rational Functions

Polnomial and Rational Functions 5 Figure 1 35-mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia

### Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

### Which of the following expressions are monomials?

9 1 Stud Guide Pages 382 387 Polnomials The epressions, 6, 5a 2, and 10cd 3 are eamples of monomials. A monomial is a number, a variable, or a product of numbers and variables. An eponents in a monomial

### Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

### k y = where k is the constant of variation and

Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.

### Syllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.

Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in

### 3.1 Exponential Functions and Their Graphs

.1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

### Attributes of Polynomial Functions VOCABULARY

8- Attributes of Polnomial Functions TEKS FCUS Etends TEKS ()(A) Graph the functions f () =, f () =, f () =, f () =, f () = b, f () =, and f () = log b () where b is,, and e, and, when applicable, analze

### f 0 ab a b: base f

Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential

### ab is shifted horizontally by h units. ab is shifted vertically by k units.

Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an

### 4Cubic. polynomials UNCORRECTED PAGE PROOFS

4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review

### Algebra Notes Quadratic Functions and Equations Unit 08

Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic

### Review Topics for MATH 1400 Elements of Calculus Table of Contents

Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

### Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions

Name Class Date -1 Finding Real Roots of Polnomial Equations Etension: Graphing Factorable Polnomial Functions Essential question: How do ou use zeros to graph polnomial functions? Video Tutor prep for

### Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

### 3.1 Graph Quadratic Functions

3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your

### KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

### Intermediate Algebra 100A Final Exam Review Fall 2007

1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,

### Solve Quadratic Equations

Skill: solve quadratic equations by factoring. Solve Quadratic Equations A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. For eample, see

### Polynomial Functions of Higher Degree

SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona

### Algebra/Trigonometry Review Notes

Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR,

### Name Please print your name as it appears on the class roster.

Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes

### MATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)

MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the

### Systems of Linear Equations: Solving by Graphing

8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

### Basic ALGEBRA 2 SUMMER PACKET

Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout

### 5. Determine the discriminant for each and describe the nature of the roots.

4. Quadratic Equations Notes Day 1 1. Solve by factoring: a. 3 16 1 b. 3 c. 8 0 d. 9 18 0. Quadratic Formula: The roots of a quadratic equation of the form A + B + C = 0 with a 0 are given by the following

### Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE

The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test

### ACCUPLACER MATH 0310

The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

### 3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

### Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.

C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each

### Exponential, Logistic, and Logarithmic Functions

CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic

### Essential Question How can you factor a polynomial completely?

REASONING ABSTRACTLY 7.8 To be proficient in math, ou need to know and flexibl use different properties of operations and objects. Factoring Polnomials Completel Essential Question How can ou factor a

### = x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

### Graphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).

Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a

### Summer 2013 Advanced Algebra & Trigonometry Course Content Teacher Edition

Summer 0 Advanced Algebra & Trigonometr Course Content Teacher Edition Algebra and Trigonometr (Third Edition) Beecher, Penna, Bittinger Addision Wesle (Februar, 007) Basic Concepts of Algebra R. The

### Name Class Date. Multiplying Two Binomials Using Algebra Tiles

Name Class Date Multiplying Polynomials Going Deeper Essential question: How do you multiply polynomials? 6-5 A monomial is a number, a variable, or the product of a number and one or more variables raised

### EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n

Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m

### Use Properties of Exponents

4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers

### Objectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation

9-6 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the

### Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution

### Pre-Algebra Notes Unit 12: Polynomials and Sequences

Pre-Algebra Notes Unit 1: Polynomials and Sequences Polynomials Syllabus Objective: (6.1) The student will write polynomials in standard form. Let s review a definition: monomial. A monomial is a number,

### Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.

Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions -: Points and Lines Sstem of Linear Equations: - two or more linear equations on the same coordinate grid. Solution of

### Solving Quadratic Equations

9 Solving Quadratic Equations 9. Properties of Radicals 9. Solving Quadratic Equations b Graphing 9. Solving Quadratic Equations Using Square Roots 9. Solving Quadratic Equations b Completing the Square

### We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:

R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)

### Sample Problems For Grade 9 Mathematics. Grade. 1. If x 3

Sample roblems For 9 Mathematics DIRECTIONS: This section provides sample mathematics problems for the 9 test forms. These problems are based on material included in the New York Cit curriculum for 8.

### MAT 1033C -- Martin-Gay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam

MAT 33C -- Martin-Ga Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

### NEXT-GENERATION Advanced Algebra and Functions

NEXT-GENERATIN Advanced Algebra and Functions Sample Questions The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunit.

### Algebra 2 Notes Powers, Roots, and Radicals Unit 07. a. Exponential equations can be solved by taking the nth

Algebra Notes Powers, Roots, and Radicals Unit 07 Exponents, Radicals, and Rational Number Exponents n th Big Idea: If b a, then b is the n root of a. This is written n a b. n is called the index, a is

### Prep for College Algebra

Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)

### Maintaining Mathematical Proficiency

Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area

### Quadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry

Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function = - -1 0 1 1 Eample 1: Make a table,

### 4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the

### N x. You should know how to decompose a rational function into partial fractions.

Section 7. Partial Fractions 0. 0 7 0 0 0 0 Solution:, 0 Equation Equation Eq. Eq. 07. nswers will var. Section 7. Partial Fractions N You should know how to decompose a rational function into partial

### MATH College Algebra Review for Test 2

MATH 4 - College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of

### 1.7 Inverse Functions

71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

### Diagnostic Tests Study Guide

California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra

### Pure Mathematics 20 Unit 1. Systems of Equations and Linear Inequalities

Pure Mathematics 0 Unit Sstems of Equations and Linear Inequalities. Eploring Ordered Pairs and Solutions. Pages 4-: ALL. Solving Sstems of Linear Equations Graphicall Pages -:,, 4, 7, 7, 9,, 7, 9,,,,

### Essential Question How can you determine whether a polynomial equation has imaginary solutions? 2 B. 4 D. 4 F.

5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.A The Fundamental Theorem of Algebra Essential Question How can ou determine whether a polnomial equation has imaginar solutions? Cubic Equations and Imaginar

### Using Intercept Form

8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of

### 25) x x + 30 x2 + 15x ) x Graph the equation. 30) y = - x - 1

Pre-AP Algebra Final Eam Review Solve. ) A stone is dropped from a tower that is feet high. The formula h = - t describes the stoneʹs height above the ground, h, in feet, t seconds after it was dropped.

### POLYNOMIALS CHAPTER 2. (A) Main Concepts and Results

CHAPTER POLYNOMIALS (A) Main Concepts and Results Meaning of a Polynomial Degree of a polynomial Coefficients Monomials, Binomials etc. Constant, Linear, Quadratic Polynomials etc. Value of a polynomial

### Lesson 10.1 Solving Quadratic Equations

Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One -intercept and all nonnegative y-values b. The verte in the third quadrant and no

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Spring 0 Math 08 Eam Preparation Ch Dressler Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the quadratic equation b the square root propert.

### Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)

Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,

### Properties of Limits

33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate

### Algebra Introduction to Polynomials

Introduction to Polynomials What is a Polynomial? A polynomial is an expression that can be written as a term or a sum of terms, each of which is the product of a scalar (the coefficient) and a series

### Properties of Rational Exponents PROPERTIES OF RATIONAL EXPONENTS AND RADICALS. =, a 0 25 º1/ =, b /3 2. b m

Page of 8. Properties of Rational Eponents What ou should learn GOAL Use properties of rational eponents to evaluate and simplif epressions. GOAL Use properties of rational eponents to solve real-life

### LESSON #17 - FACTORING COMMON CORE ALGEBRA II FACTOR TWO IMPORTANT MEANINGS

1 LESSON #17 - FACTORING COMMON CORE ALGEBRA II In the study of algebra there are certain skills that are called gateway skills because without them a student simply cannot enter into many more comple

### Dividing Polynomials

3-3 3-3 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Divide using long division. 1. 161 7 2. 12.18 2.1 23 5.8 Divide. 3. 4. 6x + 15y 3 7a 2 ab a 2x + 5y 7a b Objective

### Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

### MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS

MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,

### 15.2 Graphing Logarithmic

Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b 1 related to the graph of f () = log b? Resource Locker Eplore 1 Graphing

### ALGEBRA 1 CP FINAL EXAM REVIEW

ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8

### Nonlinear Systems. No solution One solution Two solutions. Solve the system by graphing. Check your answer.

8-10 Nonlinear Sstems CC.9-1.A.REI.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. Objective Solve sstems of equations in two

### RAMAPO&INDIAN*HILLS*SCHOOL*DISTRICT*

1 * * Algebra 2 CP Summer Packet RAMAPO&INDIAN*HILLS*SCHOOL*DISTRICT* DearRamapo*IndianHillsStudent: Pleasefindattachedthesummerpacketforourupcomingmathcourse.Thepurposeof thesummerpacketistoprovideouwithanopportunittoreviewprerequisiteskillsand

### Unit 11 - Solving Quadratic Functions PART TWO

Unit 11 - Solving Quadratic Functions PART TWO PREREQUISITE SKILLS: students should be able to add, subtract and multiply polynomials students should be able to factor polynomials students should be able

### 15.2 Graphing Logarithmic

Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > 0 and b 1 related to the graph of f () = log b? Resource Locker A.5.A Determine

### ALGEBRAIC EXPRESSIONS AND POLYNOMIALS

MODULE - ic Epressions and Polynomials ALGEBRAIC EXPRESSIONS AND POLYNOMIALS So far, you had been using arithmetical numbers, which included natural numbers, whole numbers, fractional numbers, etc. and

### Applied Algebra II Semester 2 Practice Exam A DRAFT. 6. Let f ( x) = 2x A. 47 B. 92 C. 139 D. 407

Applied Algebra II Semester Practice Eam A. Find the solution set of { + 0, 0} { + i 0, i 0} { + i, i } { +, } + = 9.. Let f ( ) = and ( ) 0 g =. Which epression is equivalent to f g? ( ) ( ). What is

### Section 3.3 Graphs of Polynomial Functions

3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section

### Chapter 3-1 Polynomials

Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents