GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function 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. Another example: x²+7 factors into. 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:
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). We extend this to the factor theorem. If the remainder after plugging in a number is zero, that means that number is a zero of the polynomial. In turn, it can be written as a factor: Example: Determine if (x 3) is a factor of the polynomial below. If (x 3) is a factor of the polynomial, that means 3 is the zero (set the factor equal to 0). So lets do synthetic division with 3. Since the remainder came out to be 0, we can say that 3 is a zero of the polynomial and (x 3) is a factor. Practice Determine if the following are factors of the polynomials. 1. Solution
2. Solution 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. Special NOTE: Around minute 6:15 the instructor incorrectly multiplies 23*9. She should get 207, but gets 107 instead. 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 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 in longer than it is wide and 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: Solving: 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 time. where h is the height and t is the 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 Polynomial Functions Quiz It is now time to complete the "Zeros of Polynomial Functions" 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. 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. 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 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.
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. 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, 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. 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. 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 Select Graphs of Polynomial Functions Project Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment when finished. A rubric is available in the sidebar.