Descartes' Rule of Signs
Descartes' Rule of Signs tells you how many positive, negative, and complex zeros exist in a given
function.
f(x)=7x510x4+3x3+4x213x+9
Positive Real Zeros
Text Version
f(x
= 7x510x4+3x3+4x213x+9
)
+
+
+
Number of Sign Changes
3
Th
The Fundamental Theorem of Algebra
Example 1
Determine the zeros of f(x)=(x13)(x18)(x+9).
Replace the function notation f(x) with zero.
f(x)=(x13)(x18)(x+9)
0=(x13)(x18)(x+9)
This function is already in factored form. Finding the zeros is as simple as set
3.04 Lesson Summary
A quadratic equation is an equation of degree 2. The graph of a quadratic equation is in the shape of
a parabola which looks like an arc. The general form of the equation is represented by
f(x) = a(x h)2 + k
The vertex, or turning poin
Rational Root Theorem
Example 1
Find all possible rational roots of f(x)=x3+7x29x+8.
Step 1: Find the factors of p.
The constant term, 8, is p. Eight is divisible by 1, 2, 4 and 8. But remember, 1, 2, 4, and 8 are
also factors.
Therefore, 1, 2, 4, and 8 a
04.03 Lesson Summary
Essential Questions
To achieve mastery of this lesson, make sure that you develop responses to the essential questions
listed below.
What key features of a polynomial can be found using the Fundamental Theorem of Algebra
and the Facto
7.03 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the essential questions
listed below.
How are exponential functions related to logarithmic functions?
How can a logarithmic function be created?
How can the
04.05 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the essential questions
listed below.
What theorems can be utilized in solving polynomials and their depressed equations?
How can polynomial functions be writt
3.07 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the essential questions
listed below.
What methods can be used to solve for quadratic equations?
How are solutions, roots, and x-intercepts of a quadratic relat
3.9 Notes
Using the distance formulas, find the equation of a parabola with a focus of (3, 4) and a directrix of y
= 2.
Check my work.
=
=
(x2 3)2 + (y2 4)2 = (y2 + 2)2.
(x2 3)2+ (y2 4)2= (y2 + 2)2
(x2 3)2+ y22 8y2 + 16 = y22 + 4y2 + 4
(x2 3)2= 12y2 12
(x
04.01 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the following essential
question.
How can long division be used to find the factors of polynomial functions?
Steps to Dividing Polynomials
The steps used in th
3.7
Solve x2 - 6x + 12 = 0 using the Quadratic Formula
Before the quadratic formula may be applied, the quadratic equation must be written in standard
form. In this case, it already is, so this step may be skipped. Also, any GCFs should be factored from
t
3.02 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the following essential
questions:
How can a Greatest Common Factor be separated from an expression?
What methods can be used to rewrite square trinomials and d
03.01 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the essential question
listed below.
How can a Greatest Common Factor be separated from an expression?
What methods can be used to rewrite square trinomials an