1. 4.1 Polynomial Functions
Objectives: Determine roots of polynomial equations. Apply the Fundamental Theorem of Algebra.

2. Polynomial in one variable:
A = P(1 + r)^t
A→final amount
P →original amount
r →rate (as a decimal)
t →time (years)
Compound Interest:
Polynomial in one variable:
an expression of the form
a0x^n + a1x^n + … + an-1x^n + an
where the coefficients a0,a1,…an represent
complex numbers, a0 is not zero, and n
represents a nonnegative integer.
ex: 10x² + 4x + 2

3. Ex. 1)
A collector of Cute Cuddlies bean bags anticipated that they would appreciate in value. With that in mind, she bought 12 for $5.00 each in 1998, 20 for $6.00 each in 1999, and 22 for $6.50 each in 2000.
a.) write a function in one variable that models the value of the collection in 2001.
b.) use the function to determine the value of the collection in 2001 if the Cuddlies appreciate at an average rate of 12% per year.

4. Degree:
Greatest exponent of a variable in a polynomial.
Leading Coefficient:
The coefficient of the variable with the greatest exponent.
Zeros/roots:
The solution for a polynomial equation. (Where a graph hits the x-axis.)
Consider the polynomial function:
f(x) = 3x^4 - x³ + x² + x – 1
a.) state the degree and leading coefficient of the polynomial.
b.) Determine whether -2 is a zero of f(x).
Ex. 2)

5. Imaginary Numbers:
i = √-1
i² = -1
i³ = - √-1 = -i
i^4 = 1
i^5 = √-1 again
Complex Numbers:
Any number that can be written in the form a+bi, where a and b are real numbers and i is the imaginary unit.
Pure imaginary→ a = 0, b ≠ 0

6. Fundamental Theorem of Algebra:
Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.
Fundamental Theorem of Algebra:
The degree of a polynomial indicates the number of possible roots of a polynomial equation.
Corollary to Fundamental Theorem of Algebra:
*Page 207 Ex.3)
a.) Write a polynomial equation of least degree with roots 2,3i, and -3i.
b.) Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis?

7. *Odd degree →must cross x-axis at least once.
Ex. 4)
State the number of complex roots of the equation 32x³ -32x² + 4x – 4=0. Then find the roots.
*Odd degree →must cross x-axis at least once.
*Even degree →may or may not cross x-axis. If
it does, it will cross an even
number of times.
Ex. 5)
State the number of complex roots of the equation x² + 16 =0. Then find the roots.
Ex. 6)