Fundamental Theorem of Algebra TS: Demonstrating understanding of concepts Warm-Up: T or F: A cubic...
-
Upload
hubert-may -
Category
Documents
-
view
214 -
download
0
Transcript of Fundamental Theorem of Algebra TS: Demonstrating understanding of concepts Warm-Up: T or F: A cubic...
Fundamental Theorem of AlgebraTS: Demonstrating understanding of concepts
Warm-Up:T or F: A cubic function has at least one real root.
T or F: A polynomial function can have no complex solutions.
T or F: A polynomial function could have only one imaginary solution.
T or F: A polynomial could have root 2 as its only irrational solution.
The Fundamental Theorem of AlgebraIf f(x) is a polynomial of degree n, where n > 0,
then f has at least one zero in the complex number system.
Linear Factorization TheoremIf f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors
f(x) = an(x – c1)(x – c2)∙∙∙(x – cn)
where c1, c2, …, cn are complex numbers.
Find a cubic polynomial with zeros of 2i and 3
Find the quartic polynomial with zeros -√2 and i, which passes
through (1, 6)
Factoring Polynomials so they are irreducable over the rationals, reals and complex zeros.
Factor each:
a) x4 – x2 – 20
Factoring Polynomials so they are irreducable over the rationals, reals and complex zeros.
Factor each:
b) x4 – 3x3 – x2 – 12x – 20 (Hint: x2 + 4 is a factor)
You Try:1) If -1 – 3i is a zero of x3 + 4x2 + 14x + 20, find the other zeros
2) Factor the following: x4 + 6x2 – 27
a) Irreducible over the rationals:
b) Irreducible over the reals:
c) Irreducible over the complex: