Given the following polynomial equations, determine all of the “POTENTIAL” rational roots based on the Rational Root Theorem and then using a syntheti...
3.3 - 1. 4.2. Zero of Polynomial Functions. Factor Theorem. Rational Zeros Theorem. Number of Zeros. Conjugate Zeros Theorem. Finding Zeros of a Polynomial Function .... Find a function defined by a polynomial of degree 3 that satisfies the given con
f(x) =anx n +an–1x n–1 +x n–2x n–2 +...+a 2x 2 +a 1x+a0 an, an–1, an–2,. . ., a2, a1, a0 an – 0 an f(x) = 4x2 – 3x – 7 4x2 – 3x – 7 = 0 x = 7
zero of f(x) = 2x3 - 5x2 + x + 2. Zeros of Polynomial. Functions. Objective: To find a polynomial with specified zeros, rational zeros, and other zeros. Introduction ... and the factors of the constant term (4, for this example). Possible rational ro
Objectives. Identify polynomial functions. Recognize characteristics of graphs of polynomial functions. Determine end behavior. Use factoring to find zeros of ...... 2 4. 11. y x. 12. y x. 13. y x. 14. y x. In Exercises 15–18, use the Leading Coeffic
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2.46 SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume fx() is a nonconstant polynomial with real coefficients written in standard form
inverse function. (p. 391). Key Vocabulary. According to the Fundamental Theorem of Algebra, every polynomial equation has at least one root. Sometimes the roots ..... Practice and Apply. State the degree and leading coefficient of each polynomial in
3 C. The Need for Large Trader Reporting . D. Relation to Consolidated Audit Trail Proposal . III. Description of Adopted Rule and Form . A. Large Traders
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RATIONAL FUNCTIONS. Chapter. 4. CHAPTER OBJECTIVES. • Determine roots of polynomial equations. (Lessons 4-1, 4-4). • Solve quadratic, rational, and radical equations and rational and radical inequalities. (Lessons 4-2, 4-6, 4-7). • Find the factors o
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Excel VBA Programming: Functions. (Dr. Tom Co, 9/14/2008). Introductory items: 1. A function a group of statements (or code) that yields a value. 2. There are
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Polynomial and Rational Functions. Section 2.1. Quadratic Functions and Models . 136. You should know the following facts about parabolas. □ is a quadratic function, and its graph is a parabola. □. If the parabola opens upward and the vertex is the p
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In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: • A polynomial of n th degree has ... There exists some theorems that are useful in determining zeros of polynomial function
1 es autor de Novum Organum, que es una crítica al Organum aristotélico. Para Aristóteles, la ciencia era un conocimiento teórico, es decir, tenía como meta la
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Sec 3.4 – Polynomial Functions Name:
Rational Root Theorem & Remainder Theorem
Rene’ Descartes is commonly credited for devising the Rational Root Theorem. The theorem states: Given a polynomial equation of the form 0=
+
+
+ … … … +
+
Any rational root of the polynomial equation must be some integer factor of divided by some integer factor of
Given the following polynomial equations, determine all of the “POTENTIAL” rational roots based on the Rational Root Theorem and then using a synthetic division to verify the most likely roots. 1.
+
− 8 − 12 = 0
2. 4
Potential Rational Roots:
− 12
+5 +6=0
Potential Rational Roots:
The Remainder Theorem suggests that if a polynomial function P(x) is divided by a linear factor (x – a) that the quotient will be a polynomial function, Q(x), with a possible constant remainder, r , which could be written out as: ( )=( − )∙ ( )+ If this seems a little complicated consider a similar statement but just using integers. For example (using the same colors to represent similar parts), ÷ = which could also be rewritten as: = ∙ +
The Remainder Theorem also leads to another important idea, The Factor Theorem. To state the Factor Theorem, we only need to evaluate P(a) from the Remainder Theorem. ( )= ( − )∙ ( )+ :Substitute “a” in for each “x” ( )=( )∙ ( )+
:Simplify (a – a) = 0
( )=
:Simplify 0∙Q(a) = 0
This is an important fact that basically states the remainder of the statement
M. Winking
Unit 3-4 page 50
( ) ÷ ( − ) is
( ).
Using the Remainder or Factor Theorem answer the following. 3. Using Synthetic Division evaluate + − 8 − 12 when x = 3
4. Use Synthetic Division to find the remainder of ( + − 8 − 12) ÷ ( − 3)
5. Using Synthetic Division evaluate (−2) given ( ) = 3 + 7 − 8 + 12.
6. Use Synthetic Division to determine the quotient of ( ) and ( ), given ( ) = 3 + 7 − 8 + 12 and ( )= +2
7. Given ( ) = ( + ) ∙ ( ) + , evaluate (− ).
8. Given (
9. Consider ( ) = 2 that ( ) = 5
10. Consider ( ) = + 3 − 2 + 4 and Justin used synthetic division to divide( + 3 − 2 + 4) ÷ ( − ). His work is partially shown below. Using this information determine ( ).
−1
( ) )
evaluate
+ 3 + 4 and
What value should be in box labeled “a”?
M. Winking
Unit 3-4 page 51a
=
( )
,
(6).
Using any available techniques determine the following (find exact answers). 11. Find all of the solutions to the polynomial equation − 3 + 6 − 12 + 8 = 0
M. Winking
12. Find all zeros of the polynomial function ( )= −4 + +8 −6
