* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 2/23/11 Lesson 2.6
Survey
Document related concepts
Eigenvalues and eigenvectors wikipedia , lookup
Cubic function wikipedia , lookup
Field (mathematics) wikipedia , lookup
Root of unity wikipedia , lookup
Quadratic equation wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Quartic function wikipedia , lookup
Gröbner basis wikipedia , lookup
Polynomial greatest common divisor wikipedia , lookup
Cayley–Hamilton theorem wikipedia , lookup
Horner's method wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
System of polynomial equations wikipedia , lookup
Polynomial ring wikipedia , lookup
Eisenstein's criterion wikipedia , lookup
Transcript
2.6--The Fundamental Theorem of Algebra What you should learn: To use the fundamental theorem of algebra to determine the number of zeros of polynomial functions, to find conjugate pairs of complex zeros, and to find all zeros of polynomials by using calculator and factoring. Why you should learn it: Finding zeros of polynomial functions is an important part of solving real-life problems. Finding where the graph crosses the x-axis has many real-life implications. For example when something would hit the ground. When we have a polynomial of degree n we have said that we can have at most ___ real zeros. This polynomial has at most ___ real zeros. EX: Zeros of Polynomial Functions a. The first degree polynomial zero: has exactly one b. Counting double roots, the seconddegree polynomial function has exactly two zeros: c. The third-degree polynomial function has exactly three zeros: d. The fourth-degree polynomial function has exactly four zeros: 1 From these examples you can see that the calculator will only show you the REAL zeros. We will have to use algebraic means to find the remaining zeros. EX: Find all zeros of Use calculator to find real zeros Use algebraic means to find remaining zeros (synthetic or polynomial division, factoring, quadratic formula) Notice that in the last example we got complex conjugate zeros. Complex zeros always come in pairs--complex conjugates EX: Find a 4th degree polynomial function with real coefficients, that has 1, 1, and 3i as zeros. . 2 EX: Find all zeros of given is a zero of f. HW: p. 266 1, 3, (9, 13find by factoring) (17, 25, use calc. to find real zeros first) 35, 37, 49, 53, 63, 66 = 12 problems 3