Section 2.2 (e-book 3.2) Polynomials Introduction: Polynomials are among the most interesting and important objects in mathematics. They have been studied for a countless number of years and there are many books and articles written about them. Polynomials are very easy to work with and they have extremely nice mathematical properties. The best comparison is that polynomials among all functions are the same as integers among all numbers. Polynomials are uniquely useful for modeling many problems which arise from natural phenomena. Definition 1: A monomial is an algebraic term consisting of a real number a times a nonnegative integer power n of a variable x. In short. The exponent n and the real number a are called the degree and the coefficient of the monomial, respectively. Example 1: Identify the degree and the coefficient of each of the following monomials Exercise 1: Show that if, then. The situation is dealt with in calculus which is beyond the scope of this course. However, we choose to avoid contradiction. With this choice, any real number becomes a polynomial of degree zero.
52 Definition 2: A polynomial is obtained by adding and/or subtracting several monomials. Example 2: Identify polynomials as well as nonpolynomials. Remark 2: It is rather clear that the sum, the difference and the product of two or more polynomials are also polynomials, however the ratio of two polynomials may not be a polynomial. Evaluating a Polynomial: If is given, then is obtained by substitution. Example 3: Given polynomials and, evaluate Remark 3: We usually write a polynomial in a descending (or ascending) order of the powers of the monomials involved. Example 4: Rewrite each polynomial in descending as well as ascending orders
Definition 3: The degree of a polynomial is the highest power (exponent) of its variable,. The leading term of a polynomial is the monomial in the polynomial with the highest power. The leading coefficient is the coefficient of the leading term. The constant term is the term without the variable (power of is zero). In general, a polynomial of degree is represented as, 53 where n is a non-negative integer and are all real numbers. In this setting, degree leading coefficient leading term constant term. Example 5: Identify and all the coefficients for the following polynomials Example 6: Find the degree, the leading term, the leading coefficient and the constant term of each polynomial: d)
54 Remark 4: A polynomial of degree zero is a constant function:. A polynomial of degree one is a linear function:. c) A polynomial of degree two is a quadratic function:. Definition 4: By a zero (root) of a polynomial we mean a value (may be real or complex) for which the polynomial, upon substitution of for, equals zero, that is to say. Example 7: Show that is a zero of polynomial, but is not a zero of. Show that the values is a zeros of. Properties of the Zeros of a Polynomial: i. A polynomial of degree has at most zeros. As example 7b above shows, some zeros may be real and some complex. Complex zeros, if any, come in conjugate pairs. Factor Theorem: A number is a zero of a polynomial if and only if is a factor of the polynomial. ii. The zeros of a polynomial can be found by setting and solving for x. This process is very easy for polynomials of degree 1, 2 and also for polynomials which are already in factored forms or can be factored completely easily. In such cases, the exact values of all the real and complex zeros can be found. iii. A real zero of a polynomial is in fact an x-intercept of the polynomial. So when factorization for solving cannot be obtained easily, the real zeros can be estimated by a graphing calculator. Although the complex zeros cannot be determined graphically, however we would be able to tell exactly how many complex zeros the polynomial has.
iv. If the number of zeros of a polynomial is less than the degree of the polynomial, then some real zeros are repeated more than one time. The number of times a real zero is repeated, say, is called the multiplicity of that zero and is a factor of the polynomial. In particular, if is an even integer, the graph of the polynomial would be tangent to the x-axis and not crossing it at. If is odd, then the graph would be tangent to x-axis while crossing it at. 55 Example 8: The following polynomials are all factor able, so find the exact values of all zeros, real and complex, and determine which zeros are x-intercept(s) and find multiplicities. Also, use graphical techniques to estimate the real zeros as well and compare them with their exact values. c) Construct a polynomial of least degree with as its real zeros with 5 having multiplicity 2. Construct one with degree 6.
Example 9: For the following polynomials, factorization cannot easily be obtained. UIse graphing techniques to estimate the real zeros as well as the number of its complex zeros. 56 f) Definition 5: A turning point of a polynomial is a point where the polynomial changes from increasing to decreasing or vice versa. For a polynomial, a turning point of a polynomial is a local minimum or a local maximum and vice versa. (Optional: Give an example of a function with a local maximum or minimum point which is not a turning point.) Properties of Graphs of Polynomials Let be a polynomial of degree n and leading coefficient. The graph of has the following properties: i. It is a continuous (one piece) and smooth (no sharp points) curve. iii. As discussed above, it has at most zeros which could all be real, all complex (only if n is even), or some real and some complex. Complex zeros come in conjugate pairs. The real zeros are the x-intercepts, iv. It has at most turning points. v. End Behavior of Polynomials: The graph increases or decreases without bound as x approaches infinity or negative infinity. This is called the end behavior (or the arm behavior) of the polynomial. This end behavior depends on the degree n and leading coefficient of the polynomial and it is summarized as following: n is even Up to the left & up to the right Down to the left & down to the right n is odd Down to the left & up to the right Up to the left & down to the right
57 Example 10: Use the table above to determine the end behavior of the polynomials below: c) d) Example 11: Explain why the following graphs do not represent graphs of polynomials?
58 Example 12: Consider the polynomial. Find the following. The degree. The leading term and coefficient. c) The possible number of real zeros. d) The possible number of turning points. e) Sketch the graph very neatly. f) Give the exact number of real zeros and complex zeros and explain your answer. g) Estimate the real zeros (x-intercepts) h) Estimate the coordinates of all turning points and classify each as a local minimum or local maximum. i) Find the intervals where the polynomial is increasing and decreasing. j) Use the table above to determine the end behavior of this polynomial.
59 Application Example 13: An open box is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines. write a polynomial function the represents the volume of the box. Determine the domain of this function. c) Estimate the value of x for which the volume is maximum. Example 14: The revenue (in millions of dollars)for a company from 2003 through 2010 can be modeled by where corresponds to 2003. Use a graphing calculator to approximate the relative exterema, as well as intervals on which the revenue function is increasing and decreasing over its domain., Use the results of part 9 to describe the compony s revenue over this period.
60 Exercise 1. Given polynomials and, evaluate c) 2. Rewrite the polynomial in descending as well as ascending orders 3. Identify and (see bottom of page 48) for the following polynomials 4. Find the degree, the leading term, the leading coefficient and the constant term of polynomial: 5. Find all zeros of each function without a calculator. 6. Use a graphing calculator to estimate the real zeros as well as the number of complex zeros of polynomial. 7. Consider the polynomial. Find the degree. Find the leading term and coefficient. c) Find the possible number of real zeros.
61 d) Find the possible number of turning points. e) Sketch the graph very neatly. f) Give the exact number of real zeros and complex zeros and explain your answer. g) Estimate the real zeros (x-intercepts) h) Estimate the coordinates of all turning points and classify each them as a local minimum or local maximum. i) Find the intervals where the polynomial is increasing and also intervals where it is decreasing j) Explain, by using the leading coefficient and the degree (see bottom of page 52), the end behavior of this polynomial. 8. The dimensions of a rectangular box are given in terms of x. Find the dimensions of the box for which the box has a maximum volume. (Use graphing method)