Download PreCalculus Fall 2016 Lesson 025 _Fundamental Theorem of Algebra

1
Lesson Plan #025
Class: Pre-calculus
Date: Monday December 7th, 2015
Topic: Fundamental Theorem of Algebra.
Aim: How do we use the Fundamental Theorem of Algebra?
Objectives:
1) Students will be able graph complex numbers.
Do Now:
HW #25:
Page 181 #’s 1-6
f ( x)  x 3  2 x 2  x  2
Given the polynomial function
1) State the degree of the polynomial function __________________
2) Express the polynomial function as a product of linear factors ______________________
3) State the zeros of the polynomial function _____________________
4) How does the number of zeros in the polynomial compare with the degree of the
of the polynomial?
Procedure:
Write the AIM and DO NOW
Get students working!
Take attendance
Give back work
Go over HW
Collect HW
From the Fundamental Theorem of Algebra, we can say that a
polynomial function of degree n has n zeros..
Irreducible
Quadratic
Assignment #1: Find the roots of the quadratic equation
f ( x)  x 2  9
Assignment #2: Complete the table
Degree Roots
Possible Combinations
1
1
1 Real Root
2
2
3
3
4
4
4 Real Roots, or 2 Real and 2 imaginary Roots, or 4 Imaginary Roots
etc
etc!
Assignment #3: Find the roots (real and imaginary) of the polynomial function and the multiplicities of
each root.
Medial Summary:
 A polynomial of degree n has n roots (Fundamental Theorem of Algebra)
 Roots may be Imaginary Numbers.
 A polynomial can be factored like: a(x-r1)(x-r2)... where r1, etc. are the roots.
 Imaginary Roots of polynomial functions with real coefficients always come in conjugate pairs.
 Multiplying an imaginary pair gives an Irreducible Quadratic.
 So a polynomial can be factored into all factors which are either:
o Linear Factors or
o Irreducible Quadratics
 Sometimes a factor appears more than once. That is its Multiplicity.
Sample Test Questions:
1) Is the following quadratic polynomial reducible or irreducible?
f ( x)  x 4  x 3
2
Together:
2)
3)
4)
5)
6)
3
On Your Own:
7)
8)
9) Given x3  2 x 2  3x  6  0 and x  3 is a root, find all the other roots.
10) Choose any method to find all the real zeroes of f (x) = x4 – 5x2 – 7.
11) Find all real roots and their multiplicity of the polynomial
12) Is the following quadratic polynomial reducible or irreducible?
13) Which of the following quadratics is irreducible?
4
If Enough Time:
1)
2)
3) Which of the following equations could be a quadratic equation with integral coefficients having roots
3  i and 3  i ?
A) x 2  9x  10  0 B) x 2  6x  10  0 C) x 2  9  0 D) x 2  9x  8  0 E) x 2  9x  8  0
4)
5)