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


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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) GridIn 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: ZeroProduct 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 realnumber 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. RealNumber 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 nonzero and constant.. Conversel, if the n th order differences of equall spaced data are nonzero 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 realworld applications. Reallife 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
32 Resources Algebra II Notes Unit Si: Polnomials Vocabular (Concept Grid) This activit can be done as a whole class activit or as a smallgroup activit. It can be done quickl as a review, or in more detail as instruction with compare/contrast capabilities. The word or phrase is written in the center bo Eamples are written in the bottom left section (Quadrant III) Each of the quadrants should include eamples, illustrations, images as appropriate Teacher ma give some, but students should see the pattern and provide input Definition Eamples Characteristics Non Eamples Non eamples are written in the bottom right section (Quadrant IV). Teacher ma give some, but students should see the pattern and provide input Characteristics are written in the top right section (Quadrant I). Teacher ma provide some, but students should see the pattern and provide input A definition is written in the upper left section (Quadrant II). As students offer definitions, it ma be necessar to add more information in the quadrants or to point to information that will focus their definition. Variations: The order of placing information in the quadrants can be changed. For eample, on new material, the definition could be given first, then mabe an eample or two, then ask students for more eamples or non eamples, then characteristics.. f 4 f 8 f Definition Characteristics Polnomial function Eamples Onl whole number eponents No negative eponents No imaginar coefficients Can be written as f()= or = Continuous graph no missing domain End point behavior Non Eamples i 1 The word or phrase can be left blank and then determined b students based on the information in the quadrants. Go through the word categorization quickl, then use the back of the paper (or other space) to epand instruction or practice if needed. Page of McDougal Littell
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