Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Big O notation wikipedia , lookup
History of the function concept wikipedia , lookup
Non-standard calculus wikipedia , lookup
Elementary mathematics wikipedia , lookup
Horner's method wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Vincent's theorem wikipedia , lookup
Riemann hypothesis wikipedia , lookup
System of polynomial equations wikipedia , lookup
171S4.4q Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra 4.4 Theorems about Zeros of Polynomial Functions Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial Division; The Remainder and Factor Theorems 4.4 Theorems about Zeros of Polynomial Functions 4.5 Rational Functions 4.6 Polynomial and Rational Inequalities The following lesson is a brief discussion of and suggestions relative to studying Chapter 4. 171Session4 171Session4 ( Package file ) To view this lesson, click the globe to the lower left. Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx March 26, 2013 • • • • • • Find a polynomial with specified zeros. For a polynomial function with integer coefficients, find the rational zeros and the other zeros, if possible. Use Descartes’ rule of signs to find information about the number of real zeros of a polynomial function with real coefficients. The Fundamental Theorem of Algebra Every polynomial function of degree Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Synthetic Division. Enter the User Name given by your instructor and use Quick Launch # 2300. n, with n ≥ 1, has at least one zero in the system of complex numbers. Oct 251:05 PM Oct 251:05 PM Zeros of Polynomial Functions with Real Coefficients The Fundamental Theorem of Algebra Example: Find a polynomial function of degree 4 having zeros 1, 2, 4i, and 4i. Solution: Such a polynomial has factors (x − 1),(x − 2), (x − 4i), and (x + 4i), so we have: Nonreal Zeros: If a complex number a + bi, b ≠ 0, is a zero of a polynomial function f(x) with real coefficients, then its conjugate, a − bi, is also a zero. (Nonreal zeros occur in conjugate pairs.) Irrational Zeros: If where a, b, and c are rational and b is not a perfect square, is a zero of a polynomial function f(x) with rational coefficients, then its conjugate is also a zero. Let an = 1: Example Suppose that a polynomial function of degree 6 with rational coefficients has 3 + 2i, 6i, and as three of its zeros. Find the other zeros. Solution: The other zeros are the conjugates of the given zeros, 3 2i, 6i, and There are no other zeros because the polynomial of degree 6 can have at most 6 zeros. Oct 251:05 PM Oct 251:05 PM Example Rational Zeros Theorem Let the polynomial function be represented as P(x) = anxn + an1xn1 + an2xn2 + ... + a1x + a0 Given f(x) = 2x3 − 3x2 − 11x + 6: a) Find the rational zeros and then the other zeros. b) Factor f(x) into linear factors. Solution: a) Because the degree of f(x) is 3, there are at most 3 distinct zeros. The possibilities for p/q are: where all the coefficients are integers. Consider a rational number denoted by p/q, where p and q are relatively prime (having no common factor besides 1 and 1). If p/q is a zero of P(x), then p is a factor of a0 and q is a factor of an. Use synthetic division to help determine the zeros. It is easier to consider the integers before the fractions. We try 1: We try 1: Since f(1) = 6, 1 is Since f(1) = 12, 1 is not a zero. not a zero. Oct 251:05 PM Oct 251:05 PM 1 171S4.4q Theorems about Zeros of Polynomial Functions March 26, 2013 Example continued Descartes’ Rule of Signs We try 3: Let P(x) be a polynomial function with real coefficients and a nonzero constant term. Since f(3) = 0, 3 is a zero. Thus x 3 is a factor. Using the results of the division above, we can express f(x) as f(x) = (x 3)(2x2 + 3x 2) . We can further factor 2x2 + 3x 2 as (2x 1)(x + 2). The rational zeros are −2, 3 and The complete factorization of f(x) is: The number of positive real zeros of P(x) is either: 1. The same as the number of variations of sign in P(x), or 2. Less than the number of variations of sign in P(x) by a positive even integer. The number of negative real zeros of P(x) is either: 3. The same as the number of variations of sign in P(x), or 4. Less than the number of variations of sign in P(x) by a positive even integer. A zero of multiplicity m must be counted m times. f(x) = (2x 1)(x 3)(x + 2) Oct 251:05 PM Example Oct 251:05 PM 342/2. Find a polynomial function of degree 3 with the given numbers as zeros: 1, 0, 4 What does Descartes’ rule of signs tell us about the number of positive real zeros and the number of negative real zeros? There are two variations of sign, so there are either two or zero positive real zeros to the equation. 342/4. Find a polynomial function of degree 3 with the given numbers as zeros: 2, i, i There are two variations of sign, so there are either two or zero negative real zeros to the equation. Total Number of Zeros (or Roots) = 4: Possible number of zeros (or roots) by kind: Positive 2 2 0 0 Negative 2 0 2 0 Nonreal 0 2 2 4 Oct 251:05 PM 342/6. Find a polynomial function of degree 3 with the given numbers as zeros: 5, √3, √3 Oct 251:26 PM 342/14. Find a polynomial function of degree 4 with 2 as a zero of multiplicity 1, 3 as a zero of multiplicity 2, and 1 as a zero of multiplicity 1. 342/8. Find a polynomial function of degree 3 with the given numbers as zeros: 4, 1 √5, 1 + √5 342/16. Find a polynomial function of degree 5 with 1/2 as a zero of multiplicity 2, 0 as a zero of multiplicity 1, and 1 as a zero of multiplicity 2. Oct 251:26 PM Oct 251:26 PM 2 171S4.4q Theorems about Zeros of Polynomial Functions 342/14. Find a polynomial function of degree 4 with 2 as a zero of multiplicity 1, 3 as a zero of multiplicity 2, and 1 as a zero of multiplicity 1. March 26, 2013 343/24. Suppose that a polynomial function of degree 4 with rational coefficients has the given numbers as zeros. Find the other zero(s): 6 5i, 1 + √7 n = 4 means that the polynomial function has 4 roots (zeros) when we include complex solutions (roots or zeros). x = 6 5i and 1 + √7 are given as two roots (zeros). Since x = 6 5i is a root, we know that the conjugate x = 6 + 5i is a root. 342/16. Find a polynomial function of degree 5 with 1/2 as a zero of multiplicity 2, 0 as a zero of multiplicity 1, and 1 as a zero of multiplicity 2. Since x = 1 + √7 is a root, we know that the conjugate x = 1 √7 is a root. Therefore, the four (4) roots (zeros) are x = 6 5i, 6 + 5i, x = 1 + √7, and x = 1 √7. If you want to write the polynomial function, use the roots (zeros) to obtain the factors. Multiply the factors to get the polynomial function as shown below. Oct 251:26 PM 343/26. Suppose that a polynomial function of degree 5 with rational coefficients has the given numbers as zeros. Find the other zero(s): 3/4, √3, 2i Oct 251:26 PM 343/29. Suppose that a polynomial function of degree 5 with rational coefficients has the given numbers as zeros. Find the other zero(s): 6, 3 + 4i, 4 √5 n = 5 means that the polynomial function has 5 roots (zeros) when we include complex solutions (roots or zeros). x = 3/4, x = √3, and x = 2i are given as three roots (zeros). x = 6, x = 3 + 4i, and x = 4 √5 are given as three roots (zeros). Since x = 3 + 4i is a root, we know that the conjugate x = 3 4i is a root. Since x = √3 is a root, we know that the conjugate Since x = 4 √5 is a root, we know that the conjugate x = √3 is a root. Since x = 2i is a root, we know that the conjugate x = 2i is a root. Therefore, the five (5) roots (zeros) are x = 4 + √5 is a root. Therefore, the five (5) roots (zeros) are n = 5 means that the polynomial function has 5 roots (zeros) when we include complex solutions (roots or zeros). x = 3/4, x = √3, x = √3, x = 2i, and x = 2i. Oct 251:26 PM 343/36. Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros: 5i x = 6, x = 3 + 4i, x = 3 4i, x = 4 √5, and x = 4 + √5. Oct 251:26 PM 343/44. Given that the polynomial function has the given zero, find the other zeros: f(x) = x3 x2 + x 1; 1 343/46. Given that the polynomial function has the given zero, find the other zeros: f(x) = x4 16; 2i Find the polynomial function with lowest degree (smallest n) that has x = 5i as a root (zero). The conjugate x = 5i is also a root (zero) of the function. Thus, we can write f(x) = (x + 5i )(x 5i ) = x2 + 25. Oct 293:33 PM Oct 251:26 PM 3 171S4.4q Theorems about Zeros of Polynomial Functions 343/47. Given that the polynomial function has the given zero, find the other zeros: f(x) = x3 6x2 + 13x 20; 4 March 26, 2013 343/50. List all possible rational zeros of the function: f(x) = x7 + 37x5 6x2 + 12 343/48. Given that the polynomial function has the given zero, find the other zeros: f(x) = x3 8; 2 343/52. List all possible rational zeros of the function: f(x) = 3x3 x2 + 6x 9 Oct 251:26 PM 343/53. List all possible rational zeros of the function: f(x) = 15x6 + 47x2 + 2 Oct 251:26 PM 343/62. For each polynomial function: a) Find the rational zeros and then the other zeros; that is, solve f(x) = 0. b) Factor f(x) into linear factors. f(x) = 2x3 + 7x2 + 2x 8 343/54. List all possible rational zeros of the function: f(x) = 10x25 + 3x17 35x + 6 Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx Other roots are (3 √41) / 4 and (3 + √41) / 4. Oct 251:26 PM 343/64. For each polynomial function: a) Find the rational zeros and then the other zeros; that is, solve f(x) = 0. b) Factor f(x) into linear factors. f(x) = 3x4 4x3 + x2 + 6x 2 Oct 251:26 PM 343/66. For each polynomial function: a) Find the rational zeros and then the other zeros; that is, solve f(x) = 0. b) Factor f(x) into linear factors. f(x) = x4 + 5x3 27x2 + 31x 10 Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx Divide 3x2 6x + 6 = 0 by 3 to yield x2 2x + 2 = 0. This gives x = 1+ i and x = 1 i to go with x = 1 and x = 1/3. Oct 251:26 PM Divide 3x2 6x + 6 = 0 by 3 to yield x2 2x + 2 = 0. This gives x = 1+ i and x = 1 i to go with x = 1 and x = 1/3. Oct 251:26 PM 4 171S4.4q Theorems about Zeros of Polynomial Functions 343/71. For each polynomial function: a) Find the rational zeros and then the other zeros; that is, solve f(x) = 0. b) Factor f(x) into linear factors. f(x) = (1/3)x3 (1/2)x2 (1/6)x + 1/6 March 26, 2013 344/74. Find only the rational zeros of the function. f(x) = x4 3x3 9x2 3x 10 Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx 344/76. Find only the rational zeros of the function. f(x) = 2x3 + 3x2 + 2x + 3 Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx Use this Excel program to calculate synthetic division problems. cfcc.edu/mathlab/syntheticdivision.xlsx The quadratic (1/3)x2 (1/3)x 1/3 = 0 multiplied by 3 yields an equivalent equation x2 x 1 = 0. This gives x = (1+ √5) / 2 and x = (1 √5) / 2 to go with x = 1/2. Oct 251:26 PM 344/92. What does Descartes rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? Q(x) = x4 2x2 + 12x 8 Oct 251:26 PM 344/98. Sketch the graph of the polynomial function. Follow the procedure outlined on p. 317. Use the rational zeros theorem when finding the zeros. f(x) = 3x3 4x2 5x + 2 344/96. What does Descartes rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? f(x) = x4 9x2 6x + 4 f(x) = +1x4 - 9x2 - 6x + 4 has two (2) sign changes. Thus, there are two or zero positive real roots (or zeros). f(-x) = +1x4 - 9x2 + 6x + 4 has two (2) sign changes. Thus, there are two or zero negative real roots (or zeros). Oct 251:26 PM 344/100. Sketch the graph of the polynomial function. Follow the procedure outlined on p. 317. Use the rational zeros theorem when finding the zeros. f(x) = 4x4 37x2 + 9 Oct 251:26 PM Oct 251:26 PM 344/98. Sketch the graph of the polynomial function. Follow the procedure outlined on p. 317. Use the rational zeros theorem when finding the zeros. f(x) = 3x3 4x2 5x + 2 Oct 251:26 PM 5 171S4.4q Theorems about Zeros of Polynomial Functions March 26, 2013 344/100. Sketch the graph of the polynomial function. Follow the procedure outlined on p. 317. Use the rational zeros theorem when finding the zeros. f(x) = 4x4 37x2 + 9 Oct 251:26 PM 6