The Remainder and Factor Theorems, algebra homework help

1. OPEN ENDED Sketch the graph of a polynomial function that has the indicated number and type of zeros. a. 3 real, 2 imaginary b. 4 real c. 2 imaginary
2. Writing in Math Use the information about medication on page 382 to explain how the roots of an equation can be used in pharmacology. Include an explanation of what the roots of this equation represent and an explanation of what the roots of this equation reveal about how often a patient should take this medication.

6-8
Roots and Zeros
|
Main Ideas
Determine the
number and type
of roots for a
polynomial equation
. Find the zeros of a
polynomial function
GET READY for the Lesson
When doctors prescribe medication, they give patients
instructions as to how much to take and how often it should be
taken. The amount of medication in your body varies with time.
Suppose the equation M(t) = 0.54 +3.543 – 100+2 + 350 models
the number of milligrams of a certain medication in the bloodstream
f hours after it has been taken. The doctor can use the roots of
this equation to determine how often the patient should take the
medication to maintain a certain concentration in the body.
Types of Roots You have already learned that a zero of a function f(x) is
any value c such that f(c) = 0. When the function is graphed, the real zeros
of the function are the x-intercepts of the graph.
KEY CONCEPT
Zeros, Factors, and Roots
Let F(x) = 0,4 + … + 0,*+ , be a polynomial function. Then the following
statements are equivalent
cis a zero of the polynomial function f(x).
• *- cis a factor of the polynomial F(x).
c is a root or solution of the polynomial equation f(x) = 0.
In addition, if c is a real number, then (CC) is an intercept of the graph of f(x).
The graph of f(x) = x4 – 5×2 + 4 is shown at the
right. The zeros of the function are -2, -1, 1, and
2. The factors of the polynomial are x + 2, +1,
x-1, and x – 2. The solutions of the equation
f(x) = 0 are -2,-1, 1, and 2. The x-intercepts of the
graph of f(x) are (-2, 0),(-1,0), (1,0), and (2,0).
Study Tip
Look Back
For review of
complex numbers,
see Lesson 5-4.
When you solve a polynomial equation with degree greater than zero, it
may have one or more real roots, or no real roots (the roots are imaginary
numbers). Since real numbers and imaginary numbers both belong to the
set of complex numbers, all polynomial equations with degree greater than
zero will have at least one root in the set of complex numbers. This is the
Fundamental Theorem of Algebra.
KEY CONCEPT
Fundamental Theorem of Algebra
Every polynomial equation with complex coordinates and degree greater than
zero has at least one root in the set of complex numbers.