Polynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions

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1 CCGPS Advanced Algebra Polynomial Functions Polynomial Functions Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the proper adjustments to dock with the station. Or you are on the design team for the US Olympic speed cycling event and you have to correct a flaw in the wheel balance of the cycles. Or you are starting your own business and want to ensure that you have maximum profit. Each of these situations involves the use of polynomials, which are mathematical expressions with many (poly ) terms. Polynomials are used to represent an amazing number of real world situations like the ones above, as well as in photography, sales, advertising, design, pollution and data analysis, to name just a few. Essential Questions What are the rules for polynomial operations? What is the Binomial Theorem? How do we interpret an expression in terms of the context? How do we apply and solve systems of equations? What do the characteristics of a graph represent? Module Minute Polynomials are used to represent situations in life; the cost for a plumbing repair is based on a service charge, the hourly rate and the cost of parts. These three things can be expressed in the terms of a polynomial. The rules for operations with polynomials are based on the rules for operations with numbers. The characteristics of the graph of a polynomial have significance in the context of the situation. The highest point of the graph could be the maximum profit, x intercept could be the break even point for advertising costs, or the slope of the graph could be the hourly rate. Algorithms like the Binomial Theorem and techniques to solve systems of equations are tools that can be used to solve problem in many types of situations. Key Words Polynomial The sum or difference of two or more monomials. Constant A term with degree 0: ie. a number alone, with no variable. Monomial An algebraic expression that is a constant, a variable, or a product of a constant and one or more variables (also called "terms") Binomial The sum or difference of two monomials 1/23

2 Trinomial The sum or difference of three monomials Degree of the Polynomial The largest sum of the exponents of one term in the polynomial Integers Positive, negative and zero whole numbers (no fractions or decimals) Like Terms Terms having the exact same variable(s) and exponent(s). Coefficient Number factor; number in front of the variable Imaginary Number A number that involves i which is. Complex Number A number with both a real and an imaginary part, in the form. Conjugate The same binomial expression with the opposite sign Greatest Common Factor Largest expression that will go into the terms evenly Zeros The roots of a function, also called solutions or x intercepts. Linear A 1 st power polynomial Quadratic A 2 nd power polynomial Cubic A 3 rd power polynomial Quartic A 4 th power polynomial Intercepts Points where a graph crosses an axis System of Equations n equations with n variables Point of Intersection The point(s) where the graphs cross. Consistent Has at least one solution Inconsistent Has no solution Domain The values for the x variable Range The values for the y variable Extrema Maximums and minimums of a graph Geometric Sequence A sequence in which every number in the sequence is equal to the previous number in the sequence multiplied by a constant number Series Any sequence of numbers written as a sum Finite Series The sum a finite amount of terms A handout of these key words and definitions is also available in the sidebar What To Expect Operations with Polynomials Handout Polynomials and Complex Numbers Quiz Pascal's Triangle Discussion Zeros of Polynomials Handout Zeros and Solving Systems Quiz Graphs and Series Quiz Graphs of Polynomial Functions Project Polynomial Functions Test To view the standards from this unit, please download the handout from the sidebar. Polynomials: Add and Subtract Let's do a quick review on what polynomials are and the types of polynomials. A constant is a number by itself. It is also a term with degree 0. This is because a number like 2 can be written as. Remember that since anything to the zero power is 1. Some examples of constants are 3, 6, 2/3, and π. A monomial is an algebraic expression that is a constant, a variable, or a product of a constant and one or more variables. Some examples of monomials are 2, x, 3x, 5xy,, and x/3. 2/23

3 A polynomial is the sum or difference of two or more monomials. Some examples of polynomials are x+2, +y 1, and 4ab 3b. Some polynomials have special names. A binomial is the sum or difference of two monomials. An easy way to remember this is 'bi' means 2, so a bicycle has two wheels. For example, x + 2 is a binomial. A trinomial is the sum or difference of three monomials. An easy way to remember this is 'tri' means 3, so a triangle has three sides. For example, 4x² 7x + 3y is a trinomial. There are no special names for polynomials with more than 3 terms. The exponents of the variables must be positive integers to be a polynomial. Since an expression with the variable in the denominator has a negative exponent it would not be a polynomial. To find the degree of the polynomial, you must find the degree of each monomial. In other words, add up the exponents of each term. The degree is determined by the exponent or sum of exponents that has the greatest value within the polynomial. Watch the video below for further explanation. To practice with polynomials and see how to write them in standard form, watch the following video. Adding and Subtracting Polynomials Polynomials can have operations performed on them just like numbers. We'll start with adding and subtracting. When adding polynomials, you will add like terms together. Remember that like terms have the exact same variable and exponent. If you know how to add polynomials, you will be able to subtract them! In adding polynomials you could add one of two ways...horizontal or vertical. The same is true for subtraction. Also, subtraction of polynomials can be illustrated as adding the opposite. Just like in adding polynomials, you must subtract like terms. Look at some examples in this video. 3/23

4 If you need more help with this, see the resources in the sidebar. For more practice, try the Quizlet in the sidebar. Polynomials: Multiply and Divide To multiply polynomials, the distributive property is used; which is...for all real numbers a, b, and c, a(b+c)=ab+ac and (b+c)a=ba+ca. This also is true for subtraction. (Remember, when multiplying like bases, you add the exponents together; multiplying 2 binomials, you distribute using a method called "FOIL"..) When you are Watch these 2 videos to review the steps for multiplying polynomials. If you need more review and practice, go to the sidebar and check out the resources. Binomial Theorem There are times when we have multiplied a binomial by a binomial. We've referred to this process as FOILing or distributing twice. What if we wanted to multiply a binomial by itself more than twice? Multiplying could take a long time. Finding would be very tedious and leave a lot of room for error. Instead, we can use the Binomial Theorem. The formula for the Binomial Theorem is shown here. The binomial theorem states that when raising a binomial to an integer power, the binomial coefficients can be calculated using the combination 4/23

5 where n is the integer power, and k is the order of the coefficient. This is expressed mathematically with the expression Basically, this says that the powers of the first term start with the power of the binomial and go down. The powers of the second term start with 0 and go up. And we use combinations to find the coefficients, which can be found using Pascal's Triangle. Watch this video to learn more. When raising a binomial to a power, the degree of each term will be the same as the power of the binomial. For example, in the problem, if you add the powers in each term to get the degree, they all come out to 6, the power of the original binomial. For more review (and a shortcut), watch the Binomial Theorem video in the sidebar. For more on the Binomial Theorem and Pascal's Triangle, go to the sidebar. Dividing Polynomials Let's start with dividing a polynomial by a monomial. The following video will show you the process. When dividing polynomials by something that isn't a monomial, we will use long division or synthetic division. Long division with polynomials is just like the long division that you learned in elementary school but now we will also be using variables. Divide the first term of the dividend by the first term of the divisor, then multiply and subtract. You will have a new polynomial to repeat the original process with. If there is a remainder other than zero, write the remainder as a fraction with the remainder as the numerator and the original divisor as the denominator. Watch this video to learn the steps and walk through some examples. Synthetic division is a method that removes the variables during the division process but puts them back at the end to recreate a polynomial expression. Instead of divide and subtract, you multiply and add. (Any remainders other than zero are treated the same way they are in long division, rewritten as a fraction with the remainder as the numerator and the divisor as the denominator.) This video will take you through the process and show you some examples. In all division, you can always check your solutions by multiplying the quotient by the divisor to get the original polynomial. If you need more practice, check out the videos in the sidebar. Now do the Self Check. Unless a method is specified, you can use either, keeping in mind synthetic division 5/23

6 works only with divisor in the form It is possible to synthetic divide if the coefficient of the x is a number other than 1. It does involve dividing with fractions, so usually long division is easier. For more on this, see "Extra" in the sidebar. Operations with Polynomials Assignment Select the "Operations with Polynomials" Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment. Factoring Polynomials Factoring Polynomials You have worked with some methods of factoring polynomials in a previous course. Here, we will review those methods and learn others, which will be used later in this module to solve equations. There are six ways to factor polynomials. 1. Greatest Common Factor (GCF) 2. Difference of Squares (Dif sq) 3. Trinomials (Tri) 4. Perfect Square Trinomials (PST) 5. Cubes (Cu) 6. Grouping (Gr) First 3 Methods The first 3 methods are the most common ones and they are ones you have seen before. Let's review them. Greatest common factor is always the first method to try. It can work on all types of polynomials. Difference of squares only works on binomials with subtract ("difference" means subtract). Trinomial factoring is used on any trinomials. Look at the Video Showcase for explanations and examples of each of these. Last 3 Methods Perfect square trinomials are used in a process called "complete the square". They can be done like regular trinomials, but there is a short cut if you wish to learn it. Factoring by cubes works on binomials that have perfect cubes for terms (either add or subtract). Grouping is a variation of GCF for 4 term polynomials. 6/23

7 Look at the Video Showcase for explanations and examples of each of these. To help you remember these, let's summarize. These are rearranged to keep binomials and trinomials together. Keep factoring until all the powers are gone or can't be factored further! For extra practice, go to Practice 1 and Practice 2 in the sidebar. (If you click "Show Related" at the bottom of these pages, there are several more factoring practices for each method.) Pascal's Triangle Discussion It is now time to complete the " Pascal's Triangle " discussion. A rubric for your discussion in located in the sidebar. In Lesson Topic 2, you learned about Pascal's Triangle. The video completes Pascal's Triangle out to the 5th row. In your post, give the next row of the triangle. The first person posting should give the 6th row, the next person gives the 7th row, and so on. Be sure to check which rows have been posted before you do your post. Next, research Pascal's Triangle. Find other ways that Pascal's triangle is used, or interesting patterns or facts about the Triangle. Come up with at least one item not given by anyone else and list the website where you found that item. Did you understand their post? Was this new information to you? Did you find the information interesting? Complex Numbers Complex numbers are numbers with both a real and an imaginary part. The standard form of a complex number is a + bi, where a is the real part and bi is the imaginary part of the complex number. These would look like 2 3i, 5 + i, or even 8i. Lets review imaginary numbers. An imaginary number involves i which is. The powers of i can be determined from combinations of i and i², as shown here. 7/23

8 Complex numbers are just like polynomials; operations, properties, factoring are done the same way. Adding and Subtracting Complex Numbers First, we need to review how to simplify radicals that have negatives, using i. Watch the following video to learn more about simplifying radicals. When adding and subtracting complex numbers, you combine like terms. The real parts are combined and the imaginary parts are combined. For example:. Watch this video for examples of addition and subtraction of complex numbers. For more practice, go to the more resources in the sidebar. Multiply and Divide complex Numbers Multiplying complex numbers is very similar to multiplying polynomials. Remember (x)(x) = x 2. The same is true with i, (i)(i) = i 2. To multiply 2 complex numbers, you will use the same process as multiplying 2 binomials. (You need to remember how to use FOIL to multiply.) But with complex numbers i 2 = 1, so the answer must be simplified more. Watch how to multiply complex numbers. Dividing complex numbers basically involves removing i from the denominator. (i is a radical and expressions must be simplified so there are no radicals in the denominator of a fraction.) To remove i from the denominator, you will often multiply by the conjugate, which is the same binomial expression with the opposite 8/23

9 sign. The video will show imaginary denominators and will explain complex denominators. For more practice, go to more resources in the sidebar. Quiz 1: Polynomials and Complex numbers It is now time to complete the "Polynomials and Complex numbers" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Zeros of Polynomial Functions Fundamental Theorem of Algebra The fundamental theorem of algebra states that every non constant single variable polynomial with complex coefficients has at least one complex root. In other words, a polynomial function of degree n has n roots (or solutions), including real and complex roots. Go to Math is Fun in the sidebar for examples. For Polynomial Functions, the zeros are the roots or solutions or x intercepts. (Each of these 4 terms refers to the same numbers for a function.) We will learn several ways to find these zeros. Solve by Factoring Lets start with solving polynomial equations by factoring. There is a very important property we will use called the Zero Product Property which states that: If ab = 0, then a = 0 or b = 0. This property works only for zero! You can't have two numbers whose product is 5 and assume that one of the numbers is 5 (could be 2.5 times 2)! Again, it's the ZERO product property. To solve these types of problems, you will first get the equation equal to 0, then factor the equation. They will be in the general form (x a)(x+b) = 0. Using the zero product property, we will take each factor and set it equal to zero to get the answers. Complex Q. F. in the sidebar reviews complex solutions with the Quadratic Formula. Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video, x² +4=0 was solved by subtracting 4 and taking the square root. It can also be solved by factoring using Difference of Squares and imaginary numbers. The factors would be, which would give the same answers as in the video. Though Difference of Squares factoring requires subtract, it can be done with add if you use imaginary numbers. 9/23

10 Another example: x²+7 factors into. Solve by Graphing A second way to solve polynomial equations is by using graphs. The solution to a polynomial equation is where the graph of the equation crosses the x axis, the x intercepts. For example, if the graph crossed the x axis at 3, x=3 would be a solution to the equation. In this graph the solution is x=3 & x=5. Conversely, if we can find the zeros from a graph, we can write the equation that made the graph. If the zeros are 3 and 5, the factors of the polynomial are (x 3)(x 5). Multiplying these out, we get x² 8x+15, which is the polynomial graphed here. (This expression could be multiplied by a constant to get another graph that has the same intercepts.) Use your calculator or CLICK HERE to use an online calculator to graph the equation and find the x intercepts. Try this equation: Note: You can use the trace, roots, or zoom features of your calculator to find the solutions if the intercepts are not integers. Applications of Zeros Remainder Theorem When dividing polynomials, you divide one polynomial (the dividend the number being divided) by another (the divisor does the dividing), and get a quotient and a remainder. The remainder can be zero, a constant, or a polynomial whose degree is less than the degree of the divisor. You can always check your answer by multiplying using the steps below: 10/23

11 The Remainder Theorem shown here says that the remainder will be the same answer as plugging the number into the polynomial for x and solving. In other words, if we divide a polynomial by x c, the remainder will be the same as f(c). Watch the video for some examples. For more information, go to Remainder Theorem in the sidebar. Rough Sketch Graphs If we have the zeros of a function, we can create a rough sketch of the graph. The zeros are the x intercepts, so we can graph those numbers on the x axis. Also, the constant in a polynomial equation is the y intercept. (The y intercept is the value where x = 0.) Polynomial graphs have basic shapes, depending on the degree, as shown in the table. Polynomial Function Degree Graph Constant 0 Horizontal line Linear 1 Line with slope Quadratic 2 Parabola Cubic 3 "S" shape curve 11/23

12 Quartic 4 "W" shaped Since the graph of a polynomial function is a continuous smooth curve, by plotting the intercepts, we can draw a rough sketch of the graph. Watch the video below to see how to do rough sketches. We will learn more about graphing in a later lesson. Example 1: A manufacturer needs a box that will hold 720 cubic inches of material. For transporting, the box needs to be 5 inches longer than it is wide and 20 inches high. What are the dimensions of the box? Solution: What we can gather from problem: Formula: Solving: width = x Length = x+5 (5 longer) Height = 20 Volume = 720 V = L W H Since x is a width, it can't be 9. So the dimensions are W = 4 in, L = 9 in and H = 20 in. Example 2: The height of an arrow shot by a 6 foot tall person is given by the function is the height and t is the time. where h 12/23

13 At what time would the arrow be able to hit a target 10 feet in the air? Solution: Put 10 in for h(x): Solve: So the arrow could hit a 10 foot target in 2 sec. or in are two times that would work.) sec. (Think about the path of the arrow and why there Zeros of Polynomials Assignment Select the Zeros of Polynomials Handout from the sidebar. Please note, the title on this worksheet reads, "Operations with Polynomials Handout"; the title instead should say, "Zeros of Polynomials Assignment". Record your answers in a separate document. Submit your completed assignment. Systems of Equations Solve Systems by Graphing A system of equations is n equations with n variables. In a previous course, you learned to solve systems of linear equations with 2 variables. We will review that here and extend it to simple polynomial equations. To solve systems by graphing, we graph the equations to find the point of intersection, the point(s) where the graphs cross. Systems are consistent if they have one or more solutions and inconsistent if they have no solution. Watch these videos to understand graphing systems of equations. For more review, check out Graphing and Graphing Video in the sidebar. 13/23

14 Solve systems by Substitution and Elimination There are 2 methods to solve systems of equations algebraically: Substitution and Elimination. Substitution Method With this method, you first isolate one of the variables, then substitute the resulting expression into the second equation and solve. Substitute that answer back into either of the original equations to find the second variable. Watch the following video to review this method. In dealing with functions, equations are often in the form y =, or f(x) =. When we use substitution for these functions, we are basically setting 2 equations equal to each other. For example: to solve the system, both f(x) and g(x) represent y. Therefore we can set the equations equal to each other: f(x)=g(x) or 2x 3=x+1. The solution of this equation will be the solution to the system, or the intersection point of the graphs of the functions. For more review, check out Substitution and Substitution Video in the sidebar. Elimination Method The second method is Elimination. With this method, the equations are added or subtracted so that one of the variables cancels out. The remaining equation can then be solved. Substitute that answer back into either of the original equations to find the second variable. Watch the video to review this method. For more review, check out Elimination and Elimination Video in the sidebar. Now that we have reviewed systems of linear equations, let's apply these same methods to non linear equations. We will work with one linear and one quadratic equation. Click here for a visualization of the graph of this type of system. Move the parabola to see the possible number of solutions. What did you find? There are 3 possible numbers of solutions for this system. It can have 2 solutions if the line crosses the parabola in 2 places, one solution if the line touches the parabola at only 1 point, or no solution if the graphs do not cross. Watch the video to see an example of solving with each method. For more review, check out More Systems in the sidebar 14/23

15 Problem Solving Systems of equation have applications in real world situations. They are often used in comparing things like the cost of different cell phone plans or the benefits of different investments. Let's look at a few examples in this Video Showcase. Now try a few on your own. Zeros and Solving Systems Quiz It is now time to complete the "Zeros and Solving Systems" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. A portion of this content is from cnx.org Graphs of Polynomial Functions What is a polynomial function? It is a function in the form where is called the leading coefficient, is called the leading term, and is called the constant. The coefficients are real numbers and the exponents are whole numbers. In order to be a polynomial function "n" must be a nonnegative integer. The degree of the polynomial function is "n" or the value of the term in the polynomial that has the highest exponent. Let's expand our table to identify the different types of polynomial functions. Polynomial Function Example Degree Leading Coefficient Graph Constant 0 2 Horizontal line Linear 1 4 Line with slope Quadratic 2 3 Parabola Cubic 3 10 Quartic 4 6 "S" shape curve "W" shaped Rollover each function to see the graph. 15/23

16 The graph of a polynomial function is continuous. It has no holes or breaks. It is also smooth, which means that it has no sharp corners. In general its domain is the set of all real numbers. Notice you can draw it without ever lifting your pencil. In a previous class, you learned that a translation was a vertical and/or horizontal shift. We can also translate the graph of a polynomial function. Get your graphing calculator (TI 83 or Ti 84) or CLICK HERE to use an online calculator and graph the following: The following chart should summarize your discoveries: To Graph Vertical Shifts Y = f(x) + k, k > 0 Y = f(x) k, k > 0 Horizontal Shifts Y = f(x + h), h > 0 Y = f(x h), h > 0 Draw by Raise graph by k units Lower graph by k units Shift graph left h units Shift graph right h units Change in function Add k to f(x) Subtract k from f(x) Replace x with (x + h) Replace x with (x h) To review more on Translations, go to the sidebar. Key Features of Polynomial Graphs There are several key features of polynomial graphs (and all graphs). The first is the domain and the range. The domain of polynomial functions is always the set of all real numbers. The range is determined by the degree. If the degree is odd, the range is the set of all real numbers. If the degree is even, the range is from the lowest point up to or from the highest point down to. 16/23

17 In the section of rough sketches of graphs, we learned the importance of intercepts. Finding the zeros of a function will give you the x intercepts and the constant term of the function is the y intercept. Other key features are extrema, which are the maximums and minimums of a graph. For a parabola, this is the vertex. The vertex of a quadratic polynomial in the form is (h, k). If the quadratic is in the form, the x coordinate of the vertex is. (Also, recall from the rough sketch lesson that the x coordinate is the number half way between the x intercepts.) Then plug that answer into the function to find the y coordinate. In polynomial functions with higher powers, use trace, zoom or zeros features of your calculator to find the maximums and minimums. Extrema are often called turning points of a graph. Functions have a beginning and an end. Functions can be graphed and/or solved algebraically to determine the end behavior. The end behavior of a function can be determined graphically by looking at the graph and viewing how the graph begins and ends. The behavior of a function can be determined algebraically by using the leading coefficient of the equation and the degree. If the leading coefficient is positive the graph ends in an upward direction (rises). If the leading coefficient is negative, the graph ends in a downward direction (falls). Now let's think about how many times a zero occurs. If is a factor of P(x) then r is a zero of multiplicity m of the function. Furthermore if m is odd, then the graph crosses the x axis at (r, 0) and if m is even then the graph is tangent to the x axis or touches the x axis at (r, 0). See the graph of the polynomial function below. The root 2 has a multiplicity of 2 and the root 3 has a multiplicity of 1. Also notice that since the factor (x 2) has 17/23

18 an even exponent the graph is tangent to the x axis at 2. Similarly, since the factor (x 3) has an odd exponent, the graph crosses the x axis at 3. In other words, the power of a factor determines if the graph goes through the x axis or bounces off the axis. Even power bounces off and odd power goes through. Click on this app to explore the graphs of polynomials up to quintic (5 th power) polynomials. Begin by resetting the coefficients all to 0. Then change the coefficients, starting with the moving up through the different terms., which is the constant, and Pay particular attention to the shapes of the graphs, the y intercept, the shifts of the graph, the x intercepts, the max/mins, and the end behavior. Be sure to try negative numbers also. Graphing by Hand and with a Calculator In order to examine their characteristics in detail so that we can find the patterns that arise in the behavior of polynomial functions, we can study some examples of polynomial functions and their graphs. Here are 8 polynomial functions and their accompanying graphs that we will use to refer back to throughout the task. Rollover the equation to view the graph! Each of these equations can be re expressed as a product of linear factors by factoring the equations, as shown to the right of the semicolon. 18/23

19 a. List the x intercepts of j(x) using the graph above. How are these intercepts related to the right of the semicolon? SOLUTION b. Why might it be useful to know the linear factors of a polynomial function? SOLUTION c. Although we will not factor higher order polynomial functions in this unit, you have factored quadratic functions in previous courses. For review, factor the following second degree polynomials, or quadratics d. Using these factors, find the roots of these three equations. SOLUTION Watch these videos to see how to graph quadratic equations by hand. (Notice in the first video, that the x and y intercepts are in the table. Recall that you can get the x intercepts by factoring the equation and that the y intercept is the constant.) e. Sketch a graph of the three quadratic equations above without using your calculator and then use your calculator to check your graphs. f. Although you will not need to be able to find all of the roots of higher order polynomials until a later unit, using what you already know, you can factor some polynomial equations and find their roots in a similar way. Try this one: What are the roots of this fifth order polynomial function? How many roots are there? Why are there not five roots since this is a fifth degree polynomial? SOLUTION g. Graph the equation in f on your calculator. Check that the intercepts match the roots. Find the approximate max and min. SOLUTION You will use the key features to sketch graphs by hand and identify graphs done on the calculator. Go to the Graphing Practice in the sidebar to practice with graphs of quadratics. For more information on end behavior, zeros and graphing go to Purple Math in the sidebar. Geometric Series A geometric sequence is a sequence in which every number in the sequence is equal to the previous 19/23

20 number in the sequence multiplied by a constant number. This means that the ratio between consecutive numbers in the geometric sequence is a constant. The ratio, r, is the constant number that each term is multiplied by. Determine the common factor, r, for the following geometric sequences: Watch the video below for an explanation on geometric sequences and a real world example. Now let's work on the concept of adding up the numbers belonging to geometric sequences. We call the sum of any sequence of numbers a series. If we only sum a finite amount of terms, we get a finite series. We use the symbol to mean the sum of the first n terms of a sequence:. A sum may be written out using the summation symbol. This symbol is sigma, which is the capital letter "S" in the Greek alphabet. It indicates that you must sum the expression to the right of it: where i is the index of the sum; m is the lower bound (or start index), shown below the summation symbol; n is the upper bound (or end index), shown above the summation symbol; are the terms of the sequence. The index i is increased from m to n in steps of 1. If we are summing from i=1 (which implies summing from the first term in a sequence), then we can use notation since they mean the same thing: or Examples 1. = = for any value x. We can write out each term of a geometric sequence in the general form: where n is the number of the term; is the nth term of the sequence; 20/23

21 is the first term; r is the common ratio (the ratio of any term to the previous term). By simply adding together the first n terms, we are actually writing out the series Equation 1 We may multiply the above equation by r on both sides, giving us Equation 2 You may notice that all the terms on the right side of Equation 1 and Equation 2 are the same, except the first and last terms. If we subtract Equation 1 from Equation 2, we are left with just Factoring both sides, we get: Dividing by (r 1) on both sides, we arrive at the general form of the sum of a finite geometric series: Watch the Video Series in the sidebar to hear an explanation of how to get this formula. We will be using this formula to find the sums of finite geometric series. For example: What is the sum of the first 23 terms of the sequence 5, 10, 20, 40, 80,...? Since n = 23 (the number of terms), = 5 (the first term) and r = 2 (the constant factor), Using a calculator, the sum is You will find that the sum of finite geometric series can be very large. If you click here and scroll down to the app, you can see graphs of geometric series. Adjust a and r to see different series. For more explanations and examples, go to Geometric Series in the sidebar. Graphs and Series Quiz It is now time to complete the "Graphs and Series" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. 21/23

22 A portion of this content is from cnx.org Module Wrap Up Module Checklist In this module you were responsible for completing the following assignments. Review Operations with Polynomials Handout Polynomials and Complex Numbers Quiz Pascal's Triangle Discussion Zeros of Polynomials Handout Zeros and Solving Systems Quiz Graphs and Series Quiz Graphs of Polynomial Functions Project Polynomial Functions Test Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and extra resources. Re watch videos and visit the extra resources in the sidebars as needed. Then, continue to the next page for your final assessment instructions. Review your key terms. Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a standardized test. Please follow the directions closely. Remember that you may have to use prior knowledge from previous units in order to answer the question correctly. If you have any questions or concerns, please contact your instructor. Final Assessments Polynomial Functions Test It is now time to complete the "Polynomial Functions" Test. Once you have completed all self checks, assignments, and the review items and feel confident in your understanding of this material, you may begin. You will have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly. Graphs of Polynomial Functions Project 22/23

23 Select Graphs of Polynomial Functions Project Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment. A rubric is available in the sidebar. 23/23