Download Section 4.1: Intro to Polynomial Functions

4.1.1 ­ PC ­ Polynomial Functions.notebook
February 10, 2016
Chapter 4: Polynomials
4­1: Polynomial Functions
Polynomial in one variable ­ expression that is the sum of single variable terms (the same variable)
f(x) = anxn + an­1 xn­1 +...+ a1x + a0
Degree ­ greatest exponent/power
Leading Coefficient ­ number in front of the variable with the greatest exponent
Zeros ­ a value of x that makes the polynomial equal zero. i.e. f(x) = 0
1. f(x) = 10x ­ 6x2 + x3 ­ 8
a) degree = b) leading coefficient = c) is 4 a zero?
Polynomial Equation ­ when the polynomial equals zero
Root ­ solution to the polynomial equation
*root & zero of a polynomial are the exact same thing
Roots and Zeros give us/come from Factors (a lower degree polynomial expression that can be divided evenly into the polynomial)
In the previous example f(x) = 10x ­ 6x2 + x3 ­ 8
• 4 is a zero
• thus 4 is a root of 0 = 10x ­ 6x2 + x3 ­ 8
• Making (x ­ 4) a factor of the polynomial
4.1.1 ­ PC ­ Polynomial Functions.notebook
February 10, 2016
Root/Zero can be an imaginary number
i = √­1
i2 = (√­1)2 = Complex Numbers
2, ­3i, 2i, 16, ∏
Real Numbers
2, 16, ∏, √3
Imaginary Numbers
­3i, 2i
Fundamental Theorem of Algebra
Every polynomial equation has at least one root in the set of complex numbers.
Corollary to the Fundamental Theorem of Algebra
Every polynomial can be written as a product of factors
F(x) = (x ­ r 1 )(x ­ r 2 )...
Every polynomial of degree n has exactly n complex roots
*roots may also have a multiplicity, meaning they repeat.
*Example: x2 ­ 4x + 4 = 0
4.1.1 ­ PC ­ Polynomial Functions.notebook
February 10, 2016
General Shapes of Polynomials
1. Linear x
2. Quadratic x2
End Behavior:
Even Functions ­ end same direction
Odd Functions ­ end opposite directions
3. Cubic x3
4. Quartic x4
5. Quintic x5
• The number of increasing/decreasing intervals matches the degree
• The number of times the graph changes direction(turning points) is n­1
Draw a quintic function with 3 real roots.
Draw a quintic function with 2 real roots
Draw a quintic function with 6 real roots
4.1.1 ­ PC ­ Polynomial Functions.notebook
Use your graphing calculator to look at each function. How many unique real roots does each function have?
1. f(x) = 9x4 ­ 35x2 ­ 4
2. f(x) = 32x3 ­ 32x2 +4x ­ 4
3. f(x) = x2 + 5
4. f(x) = ­4
Write a Polynomial of least degree by using its roots
1) roots 2, 4i and ­4i
2.) Roots 2, 3i, and ­3i
February 10, 2016