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: