Algebra II/Trig Honors Unit 2 Day 7: The Fundamental Theorem of Algebra
Objective: To classify the zeros of a polynomial function
Warm-up: Factor and solve x
3 
5 x
2 
8 x

48

0

How many distinct solutions does the equation have? What are they?

How many total solutions does the equation have?

This degree _________ polynomial has __________ solutions, and 4 is counted as a repeated solution.
The Fundamental Theorem of Algebra
Theorem: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution in the set of complex numbers (with or without the imaginary part).
Corollary: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solutions repeated three times is counted as 3 solutions, and so on.
What that means: An n -th degree polynomial h as exactly _________ zeros.
Example 1: Find the number of solutions or zeros a. How many solutions does the equation x
3 
5 x
2 
4 x

20

0 have? b. How many zeros does the function f
 x
4 
8 x
3 
18 x
2 
27 have?

Example 2: Find the zeros of a polynomial function
Find all zeros of f
 x
5 
4 x
4 
4 x
3 
10 x
2 
13 x

14
Step 1: List the possible rational zeros.
Step 2: Use synthetic division to find
a rational zero.
Step 3: Write in factored form.
Step 4: Find the complex zeros
using the quadratic formula.
Behavior Near Zeros – what to notice:

Only the real zeros from the previous example appear as x-intercepts on the graph.

The graph is tangent (just touches) the x-axis for the repeated zero x = -1, but actually crosses the x-axis for the other zero x = 2.
In general:

When the factor x
 k is raised to an odd power, ____________________________________

When the factor x
 k is raised to an even power, ___________________________________
Complex Conjugate Theorem:

If f is a polynomial function with real coefficients, and a
 bi is an imaginary zero of f , then a
 bi is also a zero of f .
Irrational Conjugates Theorem:

Suppose f is a polynomial function with rational coefficients, and a and b are rational numbers such that b is irrational. If a
 b is a zero of f, then a
 b is also a zero of f .

Example 3: Use zeros to write a polynomial function
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and
3 and 2

5 as zeros.
*You can check your answer by putting the zeros into your function. If you get zero, the function is correct.
Descartes’ Rule of Signs
Let f
 a n x n  a n

1 x n

1      a
2 x
2  a
1 x
 a
0
be a polynomial function with real coefficients.

The number of positive real zeros of f is equal to the number of changes in sign of coefficients of f
 
or is less than this by an even number.

The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f
 
or is less than this by an even number.
Example 4:
Use Descartes’ rule of signs
Determine the possible number of positive, real zeros, negative real zeros, and imaginary zeros for f
 
 x
6 
2 x
5 
3 x
4 
10 x
3 
6 x
2 
8 x

8 f
 

Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros

Example 5: Approximate real zeros
Approximate the real zeros of f
 x
6 
2 x
5 
3 x
4 
10 x
3 
6 x
2 
8 x

8 (using your calculator).
Example 6: Approximate real zeros of a polynomial model
A tachometer measure the speed (in revolutions per minute, or RPMs) at which an engine shaft rotates.
For a certain boat, the speed x of the engine shaft (in 100s of RPMs) and the speed s of the boat (in miles per house) are modeled by s
 

0 .
00547 x
3 
0 .
225 x
2 
3 .
62 x

11 .
0
What is the tachometer reading when the boat travels 15 miles per hour?
HW: Page 141 #3-48 (M3), 53-55