MAT 129 Precalculus Chapter 5 Notes Polynomial and Rational Functions David J. Gisch and Models Example: Determine which of the following are polynomial functions. For those that are, state the degree. a) 2 4 b) 4 2 7 The degree of a polynomial is the highest power of that polynomial, which is using the above notation. If is even we also say the polynomial has an even degree. If is odd we say the polynomial has an odd degree. c) 3 400 d) 20 e) 2 2 3 f) 1
The graph of polynomial functions must be smooth and continuous. Example: What happens when the degree of a power function is even? Example: What happens when the degree of a power function is odd? 2
What happens when the degree of a power function is even or odd? We can still use transformations to graph polynomial functions, to a certain degree (LOL). We can still use transformations to graph polynomial functions, to a certain degree (LOL). If a polynomial is not a power function we can still say much about it by looking at the degree. The degree and the leading coefficient tell us the end behavior. Larger exponents (powers) dominate smaller exponents when inputs get large. 1 2 3 4 5 6 1 4 9 16 25 36 1 8 27 64 125 216 1 16 81 256 625 1296 The leading term (the term with the highest power) is going to dominate and be more important than all remaining terms. 4 3 4 48 3
The Degree of a Polynomial The degree tells us how many possible times the graph touches/crosses the x-axis. It touches or crosses the -axis at most times. If it is an even degree it may not cross/touch the x-axis at all. If it is an odd degree, it must cross the x-axis at least once. The degree tells us the maximum possible turning points. The number maximum number of turning points is one less than the degree. It has at most -1 turns. Example: For each graph state the minimum possible degree of the polynomial and whether that degree is even or odd. a) 3 400 b) 2 2 3 Zeros/Roots of Polynomials Zeros/Roots of Polynomials If is a factor of a polynomial, then we say is a root of multiplicity. 4
Zeros, Turns, End, and Degree Example: Give me an equation of a polynomial with roots of 3, 4, and 2. 1 3 4 2 4 9 40 4 48 Is that the only answer????? Example: Give me an equation of a polynomial with roots of 0, 1, and 2. Example: Give me an equation of a polynomial that touches the x-axis at 4, crosses it at 3 and 5, has 5 turning points, and has an even degree. 5
Graphing Polynomials Example: Graph the polynomial function 4 3 Example: Graph the polynomial function 1 5 x 3 Example: Graph the polynomial function 3 2 3 6
Factoring (Over the Integers) If it is of the form where b is a perfect square then it can be factored as two conjugates. 9 3 3 it cannot be factored. Properties of, might or might not factor. Recall that the two factors must add to and multiply to. 12 4 3, might or might not factor. There are several methods to do this. I try quick trial and error. Factoring Quadratics Example: Factor (over the integers). a) 2 8 b) 8 2 c) 2 32 A rational function is a function of the form Where and are polynomial functions, and 0. Note: Since the function is in the form of a fraction we need to be mindful of values that make the denominator undefined. d) 5 24 7
Example: Find the domain of each rational function. a) We often have horizontal and vertical asymptotes when graphing rational functions. Recall that an asymptote is an imaginary boundary that the graph gets close to but never quite crosses. b) 1 c) 1 1 2 Types of Asymptotes Horizontal Asymptotes Vertical Asymptotes Oblique Asymptotes A rational function, in lowest terms, will have a vertical asymptote at if x is a factor of the denominator. When we say lowest terms we mean that the rational function has been simplified. 1 2 8
If a rational function does have a factor in the denominator that cancels there is a still a domain issue and it creates what we call a hole at if x is the canceled factor of the denominator. Graph the function 2 3 2 There is an asymptote at 2, and there is a hole at 3. The hole will not show up on your calculator s graph. 2 3 3 2 Graph the function 3 2 4 20 Try to factor the denominator. I don t see an easy answer, so use the quadratic equation. 4 16 4 1 20 64 0 So this means there are no REAL factors of the denominator. So no vertical asymptotes. Example: Find the vertical asymptotes. a) b) c) 9
Example: Create a rational function that has a vertical asymptote at 3and 4, and has a hole at 6. A rational function is called proper if the degree of the numerator is less than the degree of the denominator. If a rational function is proper, the line 0is a horizontal asymptote. 1 2 1 2 1 1 2 Graph the function 3 2 4 20 Try to factor the denominator. I don t see an easy answer, so use the quadratic equation. 4 16 4 1 20 64 0 So this means there are no REAL factors of the denominator. So no vertical asymptotes. But it is proper! 10
A rational function is called improper if the degree of the numerator is greater than or equal to the degree of the denominator. If a rational function is improper, we must factor the polynomial into the sum of a polynomial and rational function. Example: Find the horizontal asymptote. 3 1 8 2 4 1 Horizontal asymptote at y=2. 4 2 4 1 Example: Find the horizontal asymptote. 2 2 1 Example: Find all the asymptotes. 3 4 3 11
Example: Find all the asymptotes. 8 5 6 The Graph of a Rational Function Graphing Example: Give a possible equation for the graph. Graphing Example: Give a possible equation for the graph. 12
Graphing Example: Give a possible equation for the graph. Graphing Example: Give a possible equation for the graph. Graphing Example: Graph the rational function. 1 4 Graphing Example: Graph the rational function. 1 13
Graphing Example: Graph the rational function. 1 Graphing Example: Graph the rational function. 1 4 Graphing Example: Graph the rational function. 6 12 3 5 2 Polynomial and Rational Inequalities 14
Steps for Solving Inequalities Solving Rational Inequalities Example: Solve the inequality 4, and graph the solution set. 1. Bring all of the terms to the left so that zero is on the right. Divide by 1 if the leading coefficient is negative. 2. Find the zeros (roots). 3. Use the numbers found in 2 to separate the number line into intervals. Example 5.4.1 Continued 4. Select a number in each interval and evaluate f at that number. Solving Rational Inequalities Example: Solve the inequality and graph the solution set. 3 2 1 0 1. Bring all of the terms to the left so that zero is on the right. 2. Find the zeros (roots) of both the numerator and the denominator. 3. Use the numbers found in 2 to separate the number line into intervals. 15
Example Continued 4. Select a number in each interval and evaluate f at that number. Solving Rational Inequalities Example Solve the inequality and graph the solution set. 4 5 2 3 3, 1 1, 2 Solving Rational Inequalities Example Solve the inequality and graph the solution set. 2 15 0 The Real Zeros of a Polynomial Function 16
The Division Algorithm The Division Algorithm The division algorithm of polynomials is the same as the one for integers. The Remainder & Factor Theorems Remainder Theorem Let be a polynomial function. If is divided by, then the remainder is. Factor Theorem Let be a polynomial function. Then is a factor of if and only if 0. The Remainder Theorem For example, let s say I have a hunch that 1 is a factor of 8. In other words 8 1???? Using the remainder theorem, we calculate 1 1 8 1 8 9 Thus, as there is a remainder ( 0) we know 1 does not divide evenly (i.e. it is not a factor). 17
Factor Theorem Suppose I now try 2 as a factor of 8. In other words 8 2???? Long Division 2 0 0 8 Using the remainder theorem, we calculate 2 2 8 8 8 0 The remainder is zero, so my hunch is correct and 2 is indeed a factor. It remains for us to find (????) from above but now we know it is a factor. If there is no remainder, how do we find the remaining factor? Use long division, or Synthetic division Synthetic Division To understand synthetic division let us use the previous example of testing 2 as a factor of 8. In other words 8 2???? Remainder/Factor Theorems Example Use synthetic division to complete the following operation. 2 3 4 2 2 1 0 0 8 2 4 8 No Remainder, so it is a factor. 1 2 4 0 8 2 2 4 18
Remainder/Factor Theorems Example Determine if the following are factors of 2 4 3 If it is, follow up with synthetic division and factor it. a) 1 Remainder/Factor Theorems Example Use synthetic or long division to find all of the asymptotes of 2 5 6 3 4 b) 1 Find the Real Zeros Let be a polynomial function of degree, with 1. Then has at most real zeros. So there are at most n but how can we tell how many for sure? Descartes rule of signs. Rational root theorem. 19
Decartes Rule of Signs Remainder/Factor Theorems Example Find the number of possible real zeros of 3 4 3 2 3 Positives: 3 4 3 2 3 Negatives: 3 4 3 2 3 Remainder/Factor Theorems Example Find the number of possible real zeros of 8 18 11 Rational Root Theorem So we know many real zeros from Decartes theorem but we still don t know what they are. We use the rational root theorem for that. 20
Remainder/Factor Theorems Example Use the rational root theorem to find the rational roots of 3 4 3 2 3 1 3 0-4 3 2-1 -3 : 1, 3 : 1, 3 Candidates:,,, 1,,3 Work smarter, not harder! Remainder/Factor Theorems Example Use the rational root theorem and Descartes's theorem to find the rational roots of 8 18 11 Summary 21
Remainder/Factor Theorems Example Find the real zeros of 2 13 29 27 9. Use the factors you found to rewrite the polynomial in factored form. Factoring Polynomials Bounds on Zeros Every polynomial with real coefficients can be uniquely factored into a product of linear and/or irreducible quadratic functions. 4 2 3 10 A Polynomial, with real coefficients,of odd degree has at least one real zero. This is because the end behavior is opposite so it must cross the graph (i.e. have a REAL zero). 22
Bounds on Zeros Example Find the bounds on the zeros of 2 13 29 27 9. Bounds on Zeros Example Find the bounds on the zeros of 2 3 5 8 3 7 2 9. Intermediate Value Theorem Let be a polynomial function. If and and are of the opposite sign, then there is at least one real zero of between and. IVT Example Show that 1has a zero between 1 and 2. 23
Summary Find all of the indicated answers on your handout. Complex Zeros; Fundamental Theorem of Algebra Complex Numbers Complex Polynomials Recall that a complex number is always of the form where and are real numbers. Further, 1. Factor 3 10 4 3 2 7 4 12 8 24
Complex Polynomials Every complex polynomial function of degree 1 has at least one complex zero. Complex Polynomials Let be a polynomial whose coefficients are real numbers. If is a zero of, the complex conjugate is a zero of (i.e. they always come in pairs). Complex Polynomials Example You are told a polynomial of degree 5 whose coefficients are real numbers has the zeros 2, 3, and 2 4. a) Find the remaining two zeros. b) Use this info to write the function as a product of linear factors and/or irreducible quadratic functions. 25
Complex Polynomials Example You are told a polynomial of degree 4 whose coefficients are real numbers has the zeros 1, 1, and 4. a) Find the remaining zero(s). b) Use this info to write the function as a product of linear factors and/or irreducible quadratic functions. Summary Example Use all these resources to factor 2 8 20 26