VIEWS: 16 PAGES: 7 POSTED ON: 5/1/2011
Mathematics Clarification for Topic A3.5: Polynomial Functions Strand: A-Algebra In the middle grades, students see the progressive generalization of arithmetic to algebra. They learn symbolic manipulation skills and use them to solve equations. They study simple forms of elementary polynomial functions such as linear, quadratic, and power functions as represented by tables, graphs, symbols, and verbal descriptions. In high school, students continue to develop their “symbol sense” by examining expressions, equations, and functions, and applying algebraic properties to solve equations. They construct a conceptual framework for analyzing any function and, using this framework, they revisit the functions they have studied before in greater depth. By the end of high school, their catalog of functions will encompass linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric functions. They will be able to reason about functions and their properties and solve multi-step problems that involve both functions and equation- solving. Students will use deductive reasoning to justify algebraic processes as they solve equations and inequalities, as well as when transforming expressions. This rich learning experience in Algebra will provide opportunities for students to understand both its structure and its applicability to solving real-world problems. Students will view algebra as a tool for analyzing and describing mathematical relationships, and for modeling problems that come from the workplace, the sciences, technology, engineering, and mathematics. STANDARD: A3 – FAMILIES OF FUNCTIONS Students study the symbolic and graphical forms of each function family. By recognizing the unique characteristics of each family, students can use them as tools for solving problems or for modeling real-world situations. Topic A3.5 -1- Topic A3.5 Polynomial Functions HSCE: A3.5.1 Write the symbolic form and sketch the graph of simple polynomial functions. Clarification: HSCE: A3.5.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2. Clarification: HSCE: A3.5.3 Determine the maximum possible number of zeroes of a polynomial function and understand the relationship between the x-intercepts of the graph and the factored form of the function. Clarification: Topic A3.5 -2- Background Information, Tools, and Representations A polynomial is a sum of power functions whose exponents are whole numbers. For example, f(x) = 3x2 - 5x + 4 consists of three power functions: the first is degree two, the second is degree one, and the third is degree zero. When a polynomial is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term, the coefficient is 3 (called the leading coefficient), the variable is x, and the exponent is two. In the second term, the coefficient is -5, etc. The degree of a polynomial is the greatest degree of any one term. In the example, the polynomial has degree two. A polynomial of degree zero is a constant. Degree one polynomials are linear, degree two are quadratic, degree three are cubic, degree four are quartic and degree five are quintic. Therefore, it is suggested that this topic follow a study of quadratic functions. A polynomial with one term is a monomial, two terms a binomial, and three terms a trinomial. The zeros of a polynomial p are the values of x for which p(x) = 0. These values are also the x-intercepts, because they tell us where the graph of p crosses the x-axis. To predict the long-run behavior of a polynomial, write it in standard form. The polynomial behaves like its leading term. So connections should be made to the behavior of the power function represented by the leading term in the polynomial and the polynomial itself (see A3.4). However, to determine the zeros of a polynomial, write it in factored form, as a product of other polynomials. For example: Rewrite the third-degree polynomial u(x) = x3 – x2 – 6x as a product of linear polynomials. By factoring out an x and then factoring the quadratic, x2 – x – 6, we rewrite u(x) as u(x) =x3-x2-6x= x(x2-x -6) =x(x-3)(x-2). The advantage of the factored form is that we can easily find the zeros of the polynomial using the rule: If a*b=0, then a or b, or both, must equal 0. Not all polynomials can be factored. Assessable Content Incorporate concepts from A1 and A2 into assessments of this topic whenever possible. Assessments of this topic should include at least one situation where students are required to use polynomial functions to model real-world situations. Resources Topic A3.5 -3- Clarifying Examples and Activities HSCE: A3.5.1 Write the symbolic form and sketch the graph of simple polynomial functions. Example 1 Sketch the graph of a 4th degree polynomial (quartic) that crosses the x-axis at x= -1, 1, 3, 4 and is negative for large positive values of x. Write a possible formula for such a function. Answer: Negative for large positive values of x means that as x , y ; (as x increases, y decreases). There are infinitely many quartic polynomials that meet these criteria. Graph using the factored form y k ( x 1)( x 1)( x 3)( x 4) , where k is a constant that determines the leading coefficient of the polynomial. When k>0, as x +∞, then y +∞ (The graph is increasing, pointing upwards at either end). For example, using k=.3 the graphing calculator shows: When k<0, then as x +∞, y -∞ (The graph is decreasing, pointing downwards at either end). Using k=-.3 as one possible answer to the above question, we see: Try a different value for k. Try different x-intercepts. Compare and contrast the results. It is not necessary for students to write this polynomial in standard form, but you might want students to note that the leading term in the quartic is kx 4 and the constant term is k*1*-1*-3*-4 = -12k. Suggestions for differentiation Enrichment Use the quartic regression feature of a graphing calculator to find a 4th degree polynomial: Enter four x-intercepts in List 1 and List 2. Enter a fifth point that is not on the x-axis. Why do we need this fifth point? Discuss: how many points are needed to determine a 4th degree polynomial? Topic A3.5 -4- y 0.5 x 4 3.5 x3 5.5 x 2 3.5 x 6 Pick a fifth ordered point that is located below the x-axis. Find the new regression equation. What other polynomial regressions can be done using the calculator? HSCE: A3.5.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2. Example 1 Given a graph that has x-intercepts of -2, 0, 3, 6, what is the minimum degree possible for a function describing this graph. Scale on x- and y-axis is 1. Answer: 4th degree, because the polynomial has four x-intercepts. Example 2 What is the degree of the polynomial in the following? x(x2 – 4)(x + 3)(x3 -7x + 4) Answer: 7th degree, because when all the parts are multiplied together the leading term is x7. Suggestions for Differentiation Enrichment Discuss the graph from Example 1 under a vertical translation of 3. What happens to the x-intercepts? Introduce the concept of non-real roots. Use a CAS system to find the two non-real zeros of the polynomial. Topic A3.5 -5- HSCE: A3.5.3 Determine the maximum possible number of zeroes of a polynomial function and understand the relationship between the x-intercepts of the graph and the factored form of the function. Example 1 Rewrite the 3rd degree polynomial u(x) = x3 – x2 – 6x as a product of linear factors. By factoring out an x and then factoring the quadratic x2 – x – 6, we rewrite u(x) as u ( x) x3 x 2 6 x x( x 2 x 6) x( x 3)( x 2) The advantage of the factored form is that we can easily find the zeros of the polynomial using the rule: If a*b=0, then a, or b, or both must equal 0. Example 2 What is the maximum number of possible real zeros in the following function? f ( x) 12 x5 2 x 4 6 x3 8 x 2 9 x 4 A) 12 B) 6 C) 5 D) 4 Example 3 Given a graph that has x-intercepts of -2, 0, 3, 6, what is the minimum degree possible for a function describing this graph. Sketch a possible graph. Example 4 Given a graph with the following x-intercepts: -2, 4, 6 Write a factored form of the polynomial function that fits the given graph. Write a standard form of the polynomial function. Example 5 How many real solutions (zeros) exist for the function graphed below? A.2 B.3 C.4 D.5 Answer C (From MiCLiMB) Topic A3.5 -6- Suggestions for differentiation: Given the polynomial f (x) 3x 6 2x 5 4x 2 1 and f (1) 0 , is there reason to expect f (x) 3x 6 2x 5 4x 2 1 1 to have a solution? Explain why or why not? Enrichment An example of a polynomial with a third degree term: Suppose a square piece of tin measures 12 inches on each side. It is desired to make an open box from this material by cutting equal-sized squares from the corners and then bending up the sides (see figure below). Find a formula for the volume of the box as a function of the length x of the side of the square cut out of each corner. The volume V(x) = x(12 – 2x)2 is found by multiplying length•width•height. The formula for V(x) may be expanded to V(x) = 4x3–48x2+144x, which is a sum of power functions with non-negative exponents. For which values of x (domain) does this formula represent the volume of the box? Topic A3.5 -7-