Polynomial Functions - DR. D. Dambreville's Math...

6/19/2015 CCGPS Advanced Algebra

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CCGPS Advanced AlgebraPolynomial Functions

Polynomial FunctionsPicture yourself riding the space shuttle to theinternational space station. You will need tocalculate your speed so you can make theproper adjustments to dock with the station. Or you are on the design team for the USOlympic speed cycling event and you have tocorrect a flaw in the wheel balance of thecycles. Or you are starting your own businessand want to ensure that you have maximumprofit. Each of these situations involves theuse of polynomials, which are mathematicalexpressions with many (poly) terms.Polynomials are used to represent an amazingnumber of real world situations like the onesabove, as well as in photography, sales,advertising, design, pollution and data analysis,to name just a few.

Essential QuestionsWhat are the rules for polynomialoperations?What is the Binomial Theorem?How do we interpret an expression interms of the context?How do we apply and solve systems ofequations?What do the characteristics of a graphrepresent?

Module MinutePolynomials are used torepresent situations in life; thecost for a plumbing repair isbased on a service charge, thehourly rate and the cost ofparts. These three things can be expressed in the terms of a polynomial. The rules foroperations 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 pointof the graph could be the maximum profit, xintercept could be the breakeven point for advertising costs, orthe slope of the graph could be the hourly rate. Algorithms like the Binomial Theorem and techniques tosolve systems of equations are tools that can be used to solve problem in many types of situations.

Key WordsPolynomial 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 ormore variables (also called "terms")Binomial The sum or difference of two monomials



Trinomial The sum or difference of three monomialsDegree of the Polynomial The largest sum of the exponents of one term in the polynomialIntegers 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 signGreatest Common Factor Largest expression that will go into the terms evenlyZeros The roots of a function, also called solutions or xintercepts.Linear A 1st power polynomialQuadratic A 2nd power polynomialCubic A 3rd power polynomialQuartic A 4th power polynomialIntercepts Points where a graph crosses an axisSystem of Equations n equations with n variablesPoint of Intersection The point(s) where the graphs cross.Consistent Has at least one solutionInconsistent Has no solutionDomain The values for the x variableRange The values for the y variableExtrema Maximums and minimums of a graphGeometric Sequence A sequence in which every number in the sequence is equal to the previousnumber in the sequence multiplied by a constant numberSeries Any sequence of numbers written as a sumFinite 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 HandoutPolynomials and Complex Numbers QuizPascal's Triangle DiscussionZeros of Polynomials HandoutZeros and Solving Systems QuizGraphs and Series QuizGraphs of Polynomial Functions ProjectPolynomial Functions Test

To view the standards from this unit, please download the handout from the sidebar.

Polynomials: Add and SubtractLet'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 bewritten as . Remember that

since anything to the zero power is 1. Some examples of constants are 3, 6, 2/3, and π. Amonomial 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.



A polynomial is the sum or difference of two or more monomials. Some examples of polynomials are x+2,

+y1, 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, 4x²7x + 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 thevariable 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 upthe exponents of each term. The degree is determined by the exponent or sum of exponents that has thegreatest 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 PolynomialsPolynomials can have operations performed on them just like numbers. We'll start with adding andsubtracting.

When adding polynomials, you will add like terms together. Remember that like terms have the exact samevariable and exponent.

If you know how to add polynomials, you will be able to subtract them! In adding polynomials you could addone of two ways...horizontal or vertical. The same is true for subtraction. Also, subtraction of polynomials canbe illustrated as adding the opposite. Just like in adding polynomials, you must subtract like terms.

Look at some examples in this video.



If you need more help with this, see the resources in the sidebar. For more practice, try the Quizlet in thesidebar.

Polynomials: Multiply and DivideTo 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; .) When you aremultiplying 2 binomials, you distribute using a method called "FOIL".

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 TheoremThere are times when we have multiplied a binomial by a binomial.We've referred to this process as FOILing or distributing twice. What ifwe 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 forerror. Instead, we can use the Binomial Theorem. The formula for theBinomial Theorem is shown here.

The binomial theorem states that when raising a binomial to an integer power, the binomial coefficients can becalculated using the combination



where n is the integer power, and k is the order of the coefficient. This is expressed mathematically with theexpression

Basically, this says that the powers of the first term start with the power of the binomial and go down. Thepowers of the second term start with 0 and go up. And we use combinations to find the coefficients, which canbe 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 , ifyou 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 PolynomialsLet'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 wewill also be using variables. Divide the first term of the dividend by the first term of the divisor, then multiply andsubtract. You will have a new polynomial to repeat the original process with. If there is a remainder other thanzero, write the remainder as a fraction with the remainder as the numerator and the original divisor as thedenominator.

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 atthe end to recreate a polynomial expression. Instead of divide and subtract, you multiply and add. (Anyremainders other than zero are treated the same way they are in long division, rewritten as a fraction with theremainder 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 originalpolynomial.

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



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 involvedividing with fractions, so usually long division is easier. For more on this, see "Extra" in thesidebar.

Operations with Polynomials AssignmentSelect the "Operations with Polynomials" Handout from the sidebar. Record your answers in aseparate document. Submit your completed assignment.

Factoring PolynomialsFactoring Polynomials

You have worked with some methods of factoring polynomials in a previous course. Here, we will reviewthose 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 likeregular 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.



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 trinomialstogether.

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 ofthese pages, there are several more factoring practices for each method.)

Pascal's Triangle DiscussionIt is now time to complete the " Pascal's Triangle " discussion. A rubric for your discussion in locatedin the sidebar .

In Lesson Topic 2, you learned about Pascal's Triangle. The video completes Pascal's Triangle out tothe 5th row. In your post, give the next row of the triangle. The first person posting should give the 6th row, thenext person gives the 7th row, and so on. Be sure to check which rows have been posted before you do yourpost. Next, research Pascal's Triangle. Find other ways that Pascal's triangle is used, or interesting patterns orfacts about the Triangle. Come up with at least one item not given by anyone else and list the website whereyou found that item. Did you understand their post? Was this new information to you? Did you find theinformation interesting?

Complex NumbersComplex numbers are numbers with both a real and an imaginary part. The standard form of a complexnumber is a + bi, where a is the real part and bi is the imaginary part of the complex number. These wouldlook 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.



Complex numbers are just likepolynomials; operations,properties, factoring are donethe same way.

Adding andSubtracting ComplexNumbersFirst, we need to review how tosimplify radicals that havenegatives, using i.

Watch the following video tolearn more about simplifyingradicals.

When adding and subtractingcomplex numbers, youcombine like terms.

The real parts are combinedand the imaginary parts arecombined. 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) = x2.

The same is true with i, (i)(i) = i2. To multiply 2 complex numbers, you will use the same process as multiplying2 binomials.

(You need to remember how to use FOIL to multiply.) But with complex numbers i2 = 1, so the answer mustbe simplified more.

Watch how to multiply complex numbers.

Dividing complex numbers basically involves removing i from the denominator. (i is a radical and expressionsmust be simplified so there are no radicals in the denominator of a fraction.) To remove i from thedenominator, you will often multiply by the conjugate, which is the same binomial expression with the opposite



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 numbersIt is now time to complete the "Polynomials and Complex numbers" quiz. You will have a limitedamount of time to complete your quiz; please plan accordingly.

Zeros of Polynomial FunctionsFundamental Theorem of AlgebraThe fundamental theorem of algebra states that every nonconstant single variable polynomial withcomplex coefficients has at least one complex root. In other words, a polynomial function of degree n has nroots (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 xintercepts. (Each of these 4 terms refers tothe same numbers for a function.) We will learn several ways to find these zeros.

Solve by FactoringLets start with solving polynomial equations by factoring. There is a very important property we will use calledthe 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 thenumbers 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 willbe in the general form (xa)(x+b) = 0. Using the zero product property, we will take each factor and set it equalto 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 factoringusing 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 imaginarynumbers.



Another example: x²+7 factors into .

Solve by GraphingA second way to solve polynomial equations is by using graphs. The solution to a polynomial equation iswhere the graph of the equation crosses the xaxis, the xintercepts. For example, if the graph crossed the xaxis 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 findthe zeros from a graph, we can write the equation that made thegraph.

If the zeros are 3 and 5, the factors of the polynomial are (x3)(x5). Multiplying these out, we get x² 8x+15, which is thepolynomial graphed here. (This expression could be multiplied bya constant to get another graph that has the same intercepts.)

Use your calculator or CLICK HERE to use an online calculatorto graph the equation and find the xintercepts.

Try this equation:

Note: You can use the trace, roots, or zoom features of yourcalculator to find the solutions if the intercepts are not integers.

Applications of ZerosRemainder TheoremWhen 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 bymultiplying using the steps below:



The Remainder Theorem shown here says that the remainder will be the same answer as plugging thenumber into the polynomial for x and solving.

In other words, if we divide a polynomial by xc, 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 GraphsIf we have the zeros of a function, we can create a rough sketch of the graph. The zeros are the xintercepts,so we can graph those numbers on the x axis.

Also, the constant in a polynomial equation is the yintercept. (The yintercept is the value where x = 0.)

Polynomial graphs have basic shapes, depending on the degree, as shown in the table.

Polynomial Function Degree GraphConstant 0 Horizontal lineLinear 1 Line with slope

Quadratic 2 ParabolaCubic 3 "S" shape curve



Quartic 4 "W" shaped

Since the graph of a polynomial function is a continuous smooth curve, by plotting the intercepts, we can drawa 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 wideand 20 inches high. What are the dimensions of the box?

Solution:

What we can gather from problem:

width = x Length = x+5 (5 longer) Height = 20 Volume = 720

Formula:

V = L•W• H

Solving:

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 where his the height and t is the time.



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 sec. (Think about the path of the arrow and why thereare two times that would work.)

Zeros of Polynomials AssignmentSelect the Zeros of Polynomials Handout from the sidebar. Please note, the title on this worksheetreads, "Operations with Polynomials Handout"; the title instead should say, "Zeros of PolynomialsAssignment". Record your answers in a separate document. Submit your completed assignment.

Systems of EquationsSolve Systems by GraphingA 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 thathere 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) wherethe 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.



Solve systems by Substitution and EliminationThere are 2 methods to solve systems of equations algebraically: Substitution and Elimination.

Substitution MethodWith this method, you first isolate one of the variables, then substitute the resulting expression into the secondequation 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 thesefunctions, 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 2x3=x+1.

The solution of this equation will be the solution to the system, or the intersection point of the graphs of thefunctions.

For more review, check out Substitution and Substitution Video in the sidebar.

Elimination MethodThe second method is Elimination.

With this method, the equations are added or subtracted so that one of the variables cancels out. Theremaining equation can then be solved. Substitute that answer back into either of the original equations to findthe 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 nonlinearequations. 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 possiblenumber 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 theparabola in 2 places, one solution if the line touches the parabola at only 1 point, or no solution if the graphsdo not cross.

Watch the video to see an example of solving with each method.

For more review, check out More Systems in the sidebar



Problem SolvingSystems of equation have applications in real world situations. They are often used in comparing things likethe 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 QuizIt is now time to complete the "Zeros and Solving Systems" quiz. You will have a limited amount oftime 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 thepolynomial that has the highest exponent.

Let's expand our table to identify the different types of polynomial functions.

PolynomialFunction Example Degree Leading

Coefficient Graph

Constant 0 2 Horizontalline

Linear 1 4 Line withslope

Quadratic 2 3 Parabola

Cubic 3 10 "S" shapecurve

Quartic 4 6 "W"shaped

Rollover each function to see the graph.



The graph of a polynomial function is continuous. It has no holes or breaks. It is also smooth, which meansthat 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 avertical and/or horizontal shift. We can also translate the graph of a polynomial function.

Get your graphing calculator (TI83 or Ti84) or CLICK HERE to use an online calculator and graph thefollowing:

The following chart should summarize your discoveries:

To Graph Draw by Change infunction

Vertical Shifts Y = f(x) + k, k> 0

Raise graph by kunits Add k to f(x)

Y = f(x) k, k> 0

Lower graph by kunits

Subtract k fromf(x)

Horizontal ShiftsY = f(x + h), h> 0

Shift graph left hunits

Replace x with (x+ h)

Y = f(x h), h> 0

Shift graph right hunits

Replace x with (x h)

To review more on Translations, go to the sidebar.

Key Features of Polynomial GraphsThere 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 thedegree. If the degree is odd, the range is the set of all real numbers. If the degree is even, the range is fromthe lowest point up to ∞ or from the highest point down to ∞.



In the section of rough sketches of graphs, we learned the importance of intercepts. Finding the zeros of afunction will give you the xintercepts and the constant term of the function is the yintercept. Other keyfeatures 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 therough sketch lesson that the x coordinate is the number half way between the xintercepts.) Then plug thatanswer into the function to find the y coordinate. In polynomial functions with higher powers, use trace, zoomor zeros features of your calculator to find the maximums and minimums. Extrema are often called turningpoints of a graph.

Functions have a beginning and an end. Functions can be graphed and/or solved algebraically to determinethe end behavior. The end behavior of a function can be determined graphically by looking at the graph andviewing how the graph begins and ends. The behavior of a function can be determined algebraically by usingthe 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 ofmultiplicity m of the function. Furthermore if m is odd, then the graph crosses the xaxis at (r, 0) and if m iseven then the graph is tangent to the xaxis or touches the xaxis 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 (x2) has



an even exponent the graph is tangent to the x axis at 2. Similarly, since the factor (x3) has an odd exponent,the graph crosses the xaxis 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 (5th power) polynomials. Begin byresetting the coefficients all to 0. Then change the coefficients, starting with the , which is the constant, andmoving up through the different terms.

Pay particular attention to the shapes of the graphs, the yintercept, the shifts of the graph, the xintercepts,the max/mins, and the end behavior.

Be sure to try negative numbers also.

Graphing by Hand and with a CalculatorIn order to examine their characteristics in detail so that we can find the patterns that arise in the behavior ofpolynomial functions, we can study some examples of polynomial functions and their graphs. Here are 8polynomial 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 reexpressed as a product of linearfactors by factoring the equations, as shown to the right of the semicolon.



a. List the xintercepts of j(x) using the graph above. How are these intercepts related to the right of thesemicolon? 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 quadraticfunctions in previous courses. For review, factor the following second degree polynomials, or quadratics.

1.

2.

3.

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, thatthe x and y intercepts are in the table. Recall that you can get the x intercepts by factoring theequation 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 yourcalculator 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 fiveroots 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 approximatemax 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 endbehavior, zeros and graphing go to Purple Math in the sidebar.

Geometric SeriesA geometric sequence is a sequence in which every number in the sequence is equal to the previous



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 sumof 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 or notation since they mean the same thing:

Examples

1. = 2+4+8+16+32+64 = 126

2. 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 nthterm of the sequence;



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 firstand 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 41943035.

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 seedifferent series.

For more explanations and examples, go to Geometric Series in the sidebar.

Graphs and Series QuizIt is now time to complete the "Graphs and Series" quiz. You will have a limited amount of time to completeyour quiz; please plan accordingly.



A portion of this content is from cnx.org

Module Wrap UpModule ChecklistIn this module you were responsible for completing the following assignments.

Operations with Polynomials HandoutPolynomials and Complex Numbers QuizPascal's Triangle DiscussionZeros of Polynomials HandoutZeros and Solving Systems QuizGraphs and Series QuizGraphs of Polynomial Functions ProjectPolynomial Functions Test

ReviewNow that you have completed the initial assessments for this module, review the lesson material with thepractice activities and extra resources. Rewatch videos and visit the extra resources in the sidebars asneeded. Then, continue to the next page for your final assessment instructions.

Review your key terms.

Standardized Test PreparationThe following problems will allow you to apply what you have learned in this module to how you maysee questions asked on a standardized test. Please follow the directions closely. Remember that youmay 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 AssessmentsPolynomial Functions Test

It is now time to complete the "Polynomial Functions" Test. Once you have completed all selfchecks, 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



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

Polynomial Functions - DR. D. Dambreville's Math...

Documents

Transcript of Polynomial Functions - DR. D. Dambreville's Math...