Download 3-6 Fundamental Theorem of Algebra Day 1

3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Bellwork 10-30-14
Identify all the real roots of each equation.
1. 4x5 – 8x4 – 32x3 = 0
2. x3 –x2 + 9 = 9x
1
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Objectives
Use the Fundamental Theorem of Algebra and its corollary
to write a polynomial equation of least degree with given
roots.
Identify all of the roots of a polynomial equation.
2
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
You have learned several important properties about real roots
of polynomial equations.
You can use this information to write polynomial function when
given in zeros.
3
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Example 1: Writing Polynomial Functions
Write the simplest polynomial with roots –1, 2 , and 4.
3
If r is a zero of P(x), then x – r is a factor of P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
4
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Check It Out! Example 1a
Write the simplest polynomial function with the given zeros.
–2, 2, 4
If r is a zero of P(x), then x – r is a factor of P(x).
5
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Notice that the degree of the function in Example 1 is the same
as the number of zeros. This is true for all polynomial functions.
However, all of the zeros are not necessarily real zeros.
Polynomials functions, like quadratic functions, may have complex
zeros that are not real numbers.
6
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Using this theorem, you can write any polynomial function in
factor form.
To find all roots of a polynomial equation, you can use a
combination of the Rational Root Theorem, the Irrational Root
Theorem, and methods for finding complex roots, such as the
quadratic formula.
7
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Example 2: Finding All Roots of a Polynomial
Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots.
The polynomial is of degree 4, so there are exactly four roots for the
equation.
Step 1 Use the rational Root Theorem to identify rational roots.
Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.
8
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Example 2 Continued
Step 3 Test the possible real roots.
x4 – 3x3 + 5x2 – 27x – 36
–1
9
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Example 2 Continued
The polynomial factors into (x + 1)(x3 – 4x2 + 9x – 36) = 0.
Step 4 Solve x2 + 9 = 0 to find the remaining roots.
The fully factored form of the equation is:
The solutions are:
10
3­6 Fundamental Theorem of Algebra­ period 1.notebook
October 30, 2014
Homework: Textbook Page 193, #'s 1-5 odd and 11-19 odd
11