Using the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...\; 0 or  l (  ONLl 


 Arline Matthews
 2 years ago
 Views:
Transcription
1 Using the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...\; 0 or  l (  ONLl  Consider the function h(x) =IJ\ 48x 312x x {?\whose graph is pictured below. What do you notice about this graph that does make sense based on the equation of the function? L,,, (D lta.,', ;,, \,,,) AO ki r. r,k, ""',. rh Cov\c;. re... 'tr, ci,, s o.\.,o \ ''". 4 t ti\ =1111\' c,.y.j..,\,s \Vhat c clusion can you draw about the four roots of the function,.f{x)? Explain your reasoning...ft s J,s.e o..ll 1)k /\.co /\,,.Q_  0MJ..... \ JQ.. I,)(_ The goal of this lesson is to use the Rational Root Theorem to aid us in finding all of the roots of the function whether they be rational, irrational, or imaginary. Make a list of the rational roots that the Rational Root Theorem guarantees are possible. t, 0 ,:i \ >!. J ± ct,\ \ _ 2,..,f'\,r\  :.  ) 3,..,  I ± \, t 3 From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Divide.f{x) by one of the associated factors. Then, divide that result by the other associated factor. en  \ ':2. 2. 'i 1' i ' x. = 3 U1 \?) 3 , Of P 'o?> '\  "1 [.Q. roo1"1> 1) V\U& Cl\  I f\..cok ><') 7':. \) 'J J'3, ff.
2 Consider the function /(x) = 2x 4  x 3 + 7x 24x4 whose graph is pictured below :.....! :....!......, i : i : : t... ' + r i What do you notice about this graph that does not make sense based on the degree of the function?,1 ),s 4 'ou4 \ t'l) O p f 1 P. Ji What conclusion can you draw about the four roots of the ( " r T :   " / ( r r ' < fw1 c t ion,j(x)? Explain your reasoning. : r r r : . : : : : i S,rt..e.. roo SJ.S4. C1J\lZ. : r T r...,, j :, : n.ul u.,kt ;1.. i::::::::::r:::::::r::::::1:::::::1::: :::r:::::::1:::::::1:::::::1:::::::::1 \)s t. Make a list of the rational roots that the Rational Root Theorem guarantees are possible.?.. :. 1. \ ) t.2, t\\ +\ +.l "' ;, t.1,, i:1..  J  )'l _., From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Dividej(x) by one of the associated factors. Then, divide that result by the other associated fact9r. t.l \ 1 _ l '1  'i '1 =  i. 'f U>t"',.,. , \  m 'to Vl o..c... ol. ('Ot)  f>  g. m  g g (2: J5 lii (\,. You have just divided a degree 4 function twice so your tesuttmg polynomial 1s a C This function's roots will be the remaining two roots, which will be imaginary, ofj( Find these imaginary roots. 'X. 'l...\" s ::. 0 AA = S F i. = x= :!. 1L
3 Consider the function f(x) = 12(;;2( ; to answer the following questions. Make a chart of the possible comb ation f post'tive, n ative, zero, and imaginary roots forj(x). t( ): \ ' '\C\x)t a t:2.qx\o N 'I Po. 1. '( 'if 'I do O 'i N. 3 OC'" i \ o o t o O Graphj(x) using a graphing calculator. Then, on the set of axes to the right, draw the graph. What conclusion can you make about the four roots of the function,f(x)? Explain your reasoning. _, _21 Si ss <;  i "', po inut.. 0J..J.._ \Jt. t I 10 \l 0 \;;,, " \e., s c.ev' r c.j. Make a list of the rational roots that the Rational Root Theorem guarantees are possible. 1'. ' R. :, \ I j'. 2. J 1 c; I \ 0 :,.:t l J :t :k I ± J t ).:t f '2. ) '± 2.) i!, i J :t\, :t2.i ;I :t J t(./t\, ::1: 5 :..S. ± :±:5: ±.S:. ±10.:illt ;l) )... ) ) 12. ) J! ) From this list, which roots appear to be roots from the graph above? Perform synthetic division to verify that they are, in fact, roots. Dividej(x) by one of the associated factors. Then, divide that result by the other ajsociated factor. r=i, \ \ 1& _ 2..;  10 = \ , "t \\ \0, :). I 'A \ Ip '.. '4 'i "to L.Q. 'Ji, <. '+ 't' 'to r You have just divided a degree 4 function twice so your resulting polynomial is a C...,  9;... t \, r,;ie, ' 2'"t (oc) This function's roots will be the remaining two roots, which will be imaginary, of). Find these imagin roots. ')(. : _ b 1 J \o ':L_ 4,Q.C:.. t ><. t x+ <oo = 0,2.. l::2.. 'I.. ').. ""T S 0
4 1. g(x) = 3x 3 + 8x 2 + l3x + 6 1'.R :. :t \, :t 2., ± '>J ± 'P.R. :. ±1,  2:. 3 b ;t\' i ±.\,t, :t \, t3j ±b 3 " "T bx. q "= 0 'Lt l.x.\ : 0 X: :t ± J4 (\)[i)   im '2,(1).. ;l....,. :..± 't.j'i _  \ ± i.ff 2. h(x)=6x 4 +x 3 +4x 2 +10x+3?.. ':. '± \ J '! ; P.R. :. \L9. : .)..!\ J t1, t3 J ±'' \ " " l ".l 3 + l , ,  3 >  c. '  'J  a \:1) <c \ \0 2. \ \ m b 3 3, ta 3 '! 9  t.o ,  a lo  ')C ,x  (. : 0 '2,. "* \ ': [Q: 'I,. :  \ ':! j \ 1..\(.\)(.,)  \ '± s=j c.,) ':lo x:.,±.li3
5 Day #22 Homework Date Period  For exercises 14, list the possible rational roots of the given function. Then, find all roots, real and imaginary, of the function. :t \0 ) t. if m \ ! \ C\ I 2,. 2,. ,;i. \ l , :,. s,o 10 ( uj (. \ \ , s Gt \1 lo,1 '5 u] <o \1 c; 25 (o 15 l..q (o t,s:.. 0 )(.:\S "I..= 2. g(x)=6x., +19x llx14 ± \. ± l,! 1, t 1 "I + \ + 1 "'"l. _.;;.. .; :  J .)) 3),, t2, t 3/!:b \ 1'2 t 2., 12 tl Z t )  I  J )  J ;). ) J.) ±t,tt"t, ± \C\ lo 2S 2 l 2S  2\ \2\ 0 lo )I,. :.  1 \ \"\ _, t.\ ')(. : ,1,,_ 3. h(x) = 3x 48x 3 l2x x + 9 :t\)! 3., t,_ +\!l tj! ' ±\, t.3.. , 3J ) ' rn 3 i  \ \\ ;11 3 ' , .3 LQ 3.:2 q 0 3" ').. :: 4 "' = 3 X= J3 rn 3 \ I 0 t& 3 01
6 4. p(x) = 2x 3 +7x 2 +2x3 ;l\ ) i?,.±..,, 1 +\ +l :  ).:2.,.., \  \ 3 3 x ')(..\ :=.O (".,..  \ )( )(. + \ ) ::: \ ::..0 4:., )(:. '/" The graph of a quintic  polynomial function, p(x), is shown to the right. Use the graph to answer questions If a is the leading coefficient of the equation of p(x), is a< 0 or is a > O? Give a reason for your answer. o1a f > \', Q:wl \1> s e, o.. o. 6. How many roots of p(x) are imaginary? Give a reason for your ; t)<.)'1') i... C...J o..q. s. ' _,  :: I l r """"s i s.1+i.,..._\., _; roo s\. k.. / ii ) 4 3 '.!. :.r ; ' t 7. If c is the constant term of the equation of p(x), what is the value of c? Give a reason for your answer. \"' " \ \:.. st t.t. J '&"f\ \ lo,, c.= Is it possible that there are four sign changes in the equation of p(x)? Give a reason for your answer. \)IM. ''t)"' l.d o..., ) N"t\A \.)M. u: 1 '"',;,,t. """ o..c...j. w e&,...6 \ MUW\. ) 0<,,11 \ue. \.t.. S \,)\ E>\'f. "l: ) 4c c; \ "' )! k 4 1 oro
Skills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More information176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s
A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps
More information:i.( c t.t) ?>x ( \ )  a.;b 1 (o.. b )(a..+al,+ b:r) x x x 3x 4192x
 .. Factoring Cubic, Quartic, and Quintic Polynomials The number one rule of factoring is that before anything is done to the polynomial, the terms must be ordered from greatest to least dewee. Beyond
More informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More information24 Zeros of Polynomial Functions
Write a polynomial function of least degree with real coefficients in standard form that has the given zeros. 33. 2, 4, 3, 5 Using the Linear Factorization Theorem and the zeros 2, 4, 3, and 5, write f
More informationChapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64
Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of  2 7 and contains the point (3, 1). Use the pointslope
More informationAPPH 4200 Physics of Fluids
APPH 42 Physics of Fluids Problem Solving and Vorticity (Ch. 5) 1.!! Quick Review 2.! Vorticity 3.! Kelvin s Theorem 4.! Examples 1 How to solve fluid problems? (Like those in textbook) Ç"Tt=l I $T1P#(
More informationReview all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationUnit 3: HW3.5 Sum and Product
Unit 3: HW3.5 Sum and Product Without solving, find the sum and product of the roots of each equation. 1. x 2 8x + 7 = 0 2. 2x + 5 = x 2 3. 7x + 4 = 3x 2 4. 10x 2 = 5x  2 5. 5x 2 2x 3 4 6. 1 3 x2 3x
More information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
More informationMath 110 Midterm 1 Study Guide October 14, 2013
Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R
More informationMAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or seconddegree polynomial. Standard Form f x = ax 2 + bx + c,
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n1,, a 2, a 1, a 0 be real
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More information2.Chapter2Test, Form I SCORE
DATE 2.Chapter2Test, Form I SCORE Write the letter for the correct answer in the blank at the right of each question.. Find the domain of the relation {(, ), (, ), (2, )). Then determine whether the relation
More informationA repeated root is a root that occurs more than once in a polynomial function.
Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the xintercepts of a polynomial function. This information allows
More informationUnit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3)
Unit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Precalculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationS~ONOH II V~8391V M3I"3~ WVX3 lvnld
(0/0 g +noqo) 6 Ja+d04J (%gi moqo) L Ja+d04J +OWJOd wox3 "do::> D aq +OUUD::>+! PUD 6U!JJM PUD4 Jno" U! aq +smu H sap!s 4+oq uo 6U!+!JM 4+JM p.rooarou "gx D pamolid aq II!M noa SONOH II V8391V M3I"3 WVX3
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationTausend Und Eine Nacht
Connecticut College Digital Commons @ Connecticut College Historic Sheet Music Collection Greer Music Library 87 Tausend Und Eine Nacht Johann Strauss Follow this and additional works at: https:digitalcommonsconncolledusheetmusic
More informationHow many solutions are real? How many solutions are imaginary? What are the solutions? (List below):
1 Algebra II Chapter 5 Test Review Standards/Goals: F.IF.7.c: I can identify the degree of a polynomial function. F.1.a./A.APR.1.: I can evaluate and simplify polynomial expressions and equations. F.1.b./
More informationVera Babe!,ku Math Lecture 1. Introduction 0 9.1, 9.2, 9.3. o Syllabus
ntroduction o Syllabus 9.1, 9.2, 9.3. Vera Babe!,ku Math 112. Lecture 1 9.1 Limits. Application Preview Although everyone recognizes the value of eliminating any and all particulate pollution from smokestack
More informationMath /Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined
Math 400001/Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined using limits. As a particular case, the derivative of f(x)
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More informationCHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information. As x gets really large, the last terms drops off and f(x) ½x
PreAP Algebra 2 Unit 8 Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be
More informationPolynomial Functions
Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),
More informationAbstract Algebra: Chapters 16 and 17
Study polynomials, their factorization, and the construction of fields. Chapter 16 Polynomial Rings Notation Let R be a commutative ring. The ring of polynomials over R in the indeterminate x is the set
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 21 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationHomework 8 Solutions to Selected Problems
Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationI M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o
I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l
More informationCHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationUnit 5 Evaluation. MultipleChoice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number
Name I.D. Number Unit Evaluation Evaluation 0 Second Year Algebra (MTHH 039 09) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus, and other
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationNO) Tails 4,4 r p h
Algebra 1 10.3 and 10.4 Part 3 Worksheet Name: Hour: Solving Q adratics by Factoring and Taking Square Roots Worksheet 1. Match each grop.1 A.) its function. A. = x2 I B. f(x) = x + 4 D. f(x) = 3x2 5 E.
More information6A The language of polynomials. A Polynomial function follows the rule. Degree of a polynomial is the highest power of x with a nonzero coefficient.
Unit Mathematical Methods Chapter 6: Polynomials Objectives To add, subtract and multiply polynomials. To divide polynomials. To use the remainder theorem, factor theorem and rationalroot theorem to identify
More informationOperations w/polynomials 4.0 Class:
Exponential LAWS Review NO CALCULATORS Name: Operations w/polynomials 4.0 Class: Topic: Operations with Polynomials Date: Main Ideas: Assignment: Given: f(x) = x 2 6x 9 a) Find the yintercept, the equation
More informationl [ L&U DOK. SENTER Denne rapport tilhører Returneres etter bruk Dokument: Arkiv: Arkivstykke/Ref: ARKAS OO.S Merknad: CP0205V Plassering:
I Denne rapport thører L&U DOK. SENTER Returneres etter bruk UTLÅN FRA FJERNARKIVET. UTLÅN ID: 020752 MASKINVN 4, FORUS  ADRESSE STMA LANETAKER ER ANSVARLIG FOR RETUR AV DETTE DOKUMENTET. VENNLIGST
More informationCHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and
CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)
More informationSection 6.6 Evaluating Polynomial Functions
Name: Period: Section 6.6 Evaluating Polynomial Functions Objective(s): Use synthetic substitution to evaluate polynomials. Essential Question: Homework: Assignment 6.6. #1 5 in the homework packet. Notes:
More informationChapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 81 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
More informationAdvanced Algebra II 1 st Semester Exam Review
dname Advanced Algebra II 1 st Semester Exam Review Chapter 1A: Number Sets & Solving Equations Name the sets of numbers to which each number belongs. 1. 34 2. 525 3. 0.875 4. Solve each equation. Check
More informationnecessita d'interrogare il cielo
gigi nei necessia d'inegae i cie cic pe sax span s inuie a dispiegaa fma dea uce < affeandi ves i cen dea uce isnane " sienzi dei padi sie veic dei' anima 5 J i f H 5 f AL J) i ) L '3 J J "' U J J ö'
More informationDividing Polynomials: Remainder and Factor Theorems
Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.
More information1. Definition of a Polynomial
1. Definition of a Polynomial What is a polynomial? A polynomial P(x) is an algebraic expression of the form Degree P(x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 3 x 3 + a 2 x 2 + a 1 x + a 0 Leading
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More informationTheorems About Roots of Polynomial Equations. Rational Root Theorem
86 Theorems About Roots of Polynomial Equations TEKS FOCUS TEKS (7)(E) Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum
More informationr(j) ::::. X U.;,..;...h_D_Vl_5_ :;;2.. Name: ~s'~o=i Class; Date: ID: A
Name: ~s'~o=i Class; Date: U.;,..;...h_D_Vl_5 _ MAC 2233 Chapter 4 Review for the test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the derivative
More informationTheorems About Roots of Polynomial Equations. Theorem Rational Root Theorem
 Theorems About Roots of Polynomial Equations Content Standards N.CN.7 Solve quadratic equations with real coefﬁcients that have complex solutions. Also N.CN.8 Objectives To solve equations using the
More informationSerge Ballif January 18, 2008
ballif@math.psu.edu The Pennsylvania State University January 18, 2008 Outline Rings Division Rings Noncommutative Rings s Roots of Rings Definition A ring R is a set toger with two binary operations +
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationName: 6.4 Polynomial Functions. Polynomial in One Variable
Name: 6.4 Polynomial Functions Polynomial Functions: The expression 3r 2 3r + 1 is a in one variable since it only contains variable, r. KEY CONCEPT Polynomial in One Variable Words A polynomial of degree
More informationo C *$ go ! b», S AT? g (i * ^ fc fa fa U  S 8 += C fl o.2h 2 fl 'fl O ' 0> fl lh cvo *, &! 5 a o3 a; O g 02 QJ 01 fls g! r«'fl O fl s ccco
> p >>>> ft^. 2 Tble f Generl rdnes. t^t  +«0 P k*ph?  i t t i S ih l H ih d. * e Stf H2 t s  ^ d  'Ct? "fi p= + V t r & ^ C d Si d n. M. s  W ^ m» H ft ^.2. S'Sllpl e Cl h /~v S s, P s'l
More informationChapter 2: Polynomial and Rational Functions
Chapter 2: Polynomial and Rational Functions Section 2.1 Quadratic Functions Date: Example 1: Sketching the Graph of a Quadratic Function a) Graph f(x) = 3 1 x 2 and g(x) = x 2 on the same coordinate plane.
More informationSynthetic Division. Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z.
Synthetic Division By: Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z. What is Synthetic Division? Synthetic Division is a simpler way to divide a polynomial by a linear factor. You can consider
More informationdirect or inverse variation. Write the equation that models the relationship.
Name. Block Date Version A Algebra 2: Chapter 8 Test Review Directions #1&2: Determine if a variation relationship exists. Describe the data in the table as a direct or inverse variation. Write the equation
More informationName (print): Question 4. exercise 1.24 (compute the union, then the intersection of two sets)
MTH299  Homework 1 Question 1. exercise 1.10 (compute the cardinality of a handful of finite sets) Solution. Write your answer here. Question 2. exercise 1.20 (compute the union of two sets) Question
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationLesson 19 Factoring Polynomials
Fast Five Lesson 19 Factoring Polynomials Factor the number 38,754 (NO CALCULATOR) Divide 72,765 by 38 (NO CALCULATOR) Math 2 Honors  Santowski How would you know if 145 was a factor of 14,436,705? What
More information2.5 Complex Zeros and the Fundamental Theorem of Algebra
210 CHAPTER 2 Polynomial, Power, and Rational Functions What you ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients... and why These topics provide the complete
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials Have nonnegative exponents Variables ONLY in General Form n ax + a x +... + ax + ax+ a n n 1
More informationPreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
More informationRewriting Absolute Value Functions as Piecewise Defined Functions
Rewriting Absolute Value Functions as Piecewise Defined Functions Consider the absolute value function f ( x) = 2x+ 43. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
More informationEnhanced Instructional Transition Guide
11 Enhanced Instructional Transition Guide High School Courses Unit Number: 7 /Mathematics Suggested Duration: 9 days Unit 7: Polynomial Functions and Applications (15 days) Possible Lesson 1 (6 days)
More informationProperties of Graphs of Polynomial Functions Terminology Associated with Graphs of Polynomial Functions
Properties of Graphs of Polynomial Functions Terminology Associated with Graphs of Polynomial Functions Detennine what types of polynomial functions/, g, and hare graphed below Give a reason for your conclusions
More informationMath 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More information5 s. 00 S aaaog. 3s a o. gg pq ficfi^pq. So c o. H «o3 g gpq ^fi^ s 03 co *«10 eo 5^  3 d s3.s. as fe«jo. Table of General Ordinances.
5 s Tble f Generl rinnes. q=! jj 3 ri j s 3s m s3 0,0 0) fife s fert " 7 CN il rl  p D fife s 3 Ph' h ^q 3 3 (j; fe QtL. S &&X* «««i s PI 0) g #r
More informationProblem 1. CS205 Homework #2 Solutions. Solution
CS205 Homework #2 s Problem 1 [Heath 3.29, page 152] Let v be a nonzero nvector. The hyperplane normal to v is the (n1)dimensional subspace of all vectors z such that v T z = 0. A reflector is a linear
More informationMidterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k.
Name: Class: Date: ID: A Midterm Review Short Answer 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. a) b) c) 2. Determine the domain and range of each function.
More informationA. Graph the parabola. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say?
Hart Interactive Honors Algebra 1 Lesson 6 M4+ Opening Exercises 1. Watch the YouTube video Imaginary Numbers Are Real [Part1: Introduction] by Welch Labs (https://www.youtube.com/watch?v=t647cgsuovu).
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More informationAlgebra 2 Notes AII.7 Polynomials Part 2
Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Name: Date: Block: Zeros of a Polynomial Function So far: o If we are given a zero (or factor or solution) of a polynomial function, we can use division
More informationExam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus I January 28, 2016 Name: Exam 1 1. Evaluate the it x 1 (2x + 1) 2 9. x 1 (2x + 1) 2 9 4x 2 + 4x + 1 9 = 4x 2 + 4x 8 = 4(x 1)(x
More informationADDITONAL MATHEMATICS
ADDITONAL MATHEMATICS 2002 2011 CLASSIFIED REMAINDER THEOREM Compiled & Edited By Dr. Eltayeb Abdul Rhman www.drtayeb.tk First Edition 2011 7 5 (i) Show that 2x 1 is a factor of 2x 3 5x 2 + 10x 4. [2]
More informationFuture SelfGuides. E,.?, :0...,0 Q., 5...q ',D5', 4,] 1}., d'.4.., _. ZoltAn Dbrnyei Introduction. u u rt 5,4) ,,4, a. a aci,, u 4.
te SelfGi ZltAn Dbnyei Intdtin ; ) Q) 4 t? ) t _ 4 73 y S _ E _ p p 4 t t 4) 1_ ::_ J 1 `i () L VI O I4 " " 1 D 4 L e Q) 1 k) QJ 7 j ZS _Le t 1 ej!2 i1 L 77 7 G (4) 4 6 t (1 ;7 bb F) t f; n (i M Q) 7S
More informationUnit 1: Polynomial Functions SuggestedTime:14 hours
Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationSemester Review Packet
MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree
More informationHPreCalculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.
HPreCalculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify
More information1) Synthetic Division: The Process. (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem
J.F. Antona 1 Maths Dep. I.E.S. Jovellanos 1) Synthetic Division: The Process (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem 1) Synthetic division. Ruffini s rule Synthetic division (Ruffini s
More informationGeorgia Department of Education Common Core Georgia Performance Standards Framework CCGPS Advanced Algebra Unit 2
Polynomials Patterns Task 1. To get an idea of what polynomial functions look like, we can graph the first through fifth degree polynomials with leading coefficients of 1. For each polynomial function,
More information(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)
1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
67 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Identify all the real roots of each equation. 1. x 3 7x 2 + 8x + 16 = 0 1, 4 2. 2x 3 14x 12 = 0 1, 2, 3 3. x 4 + x 3 25x 2 27x = 0 4. x 4 26x 2 + 25
More informationCC Algebra Quadratic Functions Test Review. 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1?
Name: CC Algebra Quadratic Functions Test Review Date: 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1? a. c. c. b. d. Which statement best describes
More informationChapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions
Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationFoundations of Math II Unit 5: Solving Equations
Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following
More informationQuestion 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case.
Class X  NCERT Maths EXERCISE NO:.1 Question 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) (iv) (v)
More informationUse the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (12) 4 3 2
Name: Math 114 Activity 1(Due by EOC Apr. 17) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please
More informationNote: The actual exam will consist of 20 multiple choice questions and 6 showyourwork questions. Extra questions are provided for practice.
College Algebra  Unit 2 Exam  Practice Test Note: The actual exam will consist of 20 multiple choice questions and 6 showyourwork questions. Extra questions are provided for practice. MULTIPLE CHOICE.
More informationRoots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
More information3 Polynomial and Rational Functions
3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,
More informationDrury&Wliitson. Economical. Planning of Buildings. .(Chilecture B. S. DNJVERSITT' OF. 11,1. 1 ibkahy
Drury&Wliitson Economical Planning of Buildings.(Chilecture B. S. 902 DJVERSTT' OF,. ibkahy 4 f ^ ^ J' if 4 ^ A 4. T? 4'tariung iint) 4':>bor. f LBRARY or TMl University of llinois. CLASS. BOOK. VO.UMK.
More information