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CP Algebra II
Standards
Date
PD
6.7 Fundamental Theorem of Algebra
6.8 Analyzing Polynomial Graphs
State Standard: 2.8.A2.B: Solve/graph polynomial functions
Objective
At the end of this lesson, you should be able to solve and graph polynomial equations.
Daily Question:
Solve 𝑥 2 − 2𝑥 + 10 = 0 using an appropriate method.
The Fundamental Theorem of Algebra
If 𝑓(𝑥)is a polynomial function of degree n, and n > 0, then the function will contain exactly n roots. This is true
as long as we take into account that some roots are counted more than once (a double or triple root) and some roots
are imaginary.
Zeros, Factors, Solutions, and Intercepts
For a standard polynomial function 𝑓(𝑥), the following statements are equivalent.
ZERO: k is a zero of the polynomial function f.
FACTOR: x – k is a factor of the polynomial𝑓(𝑥).
SOLUTION: k is a solution of the polynomial equation 𝑓(𝑥) = 0.
If k is a real number, then the following is also equivalent.
X-INTERCEPT: k is an x-intercept of the graph of the polynomial function f.
Turning Points of Polynomial Functions
The graph of every polynomial function of degree n has AT MOST n – 1 turning points. Moreover, if a
polynomial function has n distinct REAL zeros, then its graph will have EXACTLY n – 1 turning points.
*Use a graphing calculator to find the turning points which correspond to local maximum and local minimum
values of the graph.
Directions: Determine the number of roots in the following functions, find all the zeros of the function,
determine the end behavior of the graph, and graph the functions with real roots.
1.
𝑓(𝑥) = 𝑥 4 + 5𝑥 3 + 5𝑥 2 − 5𝑥 − 6
2.
𝑓(𝑥) = 𝑥 3 − 𝑥 2 + 25𝑥 − 25
3.
𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 2𝑥 − 6
4.
𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 12𝑥 − 10
Directions: Find a polynomial equation of least degree that has real coefficients, the given zeros, and a leading
coefficient of 1.
5.
𝑥 = {2, 3, −1}
6.
𝑥 = {2, −3𝑖} *2 is a double route (d.r.)
7.
𝑥 = { ±√2, 1 + 2𝑖}