:i.( c -t.t) -?>x ( -\- ) - a.;-b 1 (o..- b )(a..+al,-+ b:r) x x x -3x 4-192x
|
|
- Scot Andrews
- 5 years ago
- Views:
Transcription
1 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 that, there is a series of things to check for to break the pol yn omial down into prime factors. 1. "Undistribute" the greatest common factoiliom.. each tenn, if possible. If the leading coefficient i-s negatiye, factor out the opposite of the greatest common factor. 2. If.the polynomial is a.binomial, check.to see it is a difference of two perfect sguares. 0-fl._ b : ( -\-'o )( _ b) 2.x3-32x 12x - 3x 3-3 1, 'lx l.)( (x-+41\ ;( ) -3x( 1-_ ) -?>x ( -\- ) - ')( ( '2- -, 6) 2x4-32.'2- ( 4 -Ho) :i.( c -t.t) 2( "t ) + 2.) - 3. lf the polynomial is a binomial, check to see if it is a sum or difference of two perfect cu bes. ' 'It '1, '=- 4, 12..,;., :2. \ '- ().. wt b J: (Q.,1 b")( a... '1.-a.,\, t b :J. J a.;-b 1 (o..- b )(a..+al,-+ b:r) x x x -3x 4-192x ( -?>)( ::i. +3x + ) x ( -!x( """ <o4) 'll. '!. ) (..\- )( -:«-3x.( +4\ )( "-4'xrtu.) 4. llthe polynomial is a trinomia!, check to see if it :will factor as two different binomials. 2x 4-13x (Q-,.."- 5) (x 'J. - 't) (-x 2._ \ ) (x 5 ) (2. -,..'l. - s) ( +2.) -, (. -t-l)(x-i)(x "- -+&S) 2x 5-10.x3 + 8x x( 'i_ 'S ) ':2 '>< ()( a - \ ) (x - 4: x (x H )( -()(x-+t()-.:2) 5. If the polyno1nial has an even a1nount of tenns, try to factor by grouping.
2 Factoring by Grouping: Often, conventional methods of factoring are not useful. In cases where the pol yn omial contains an even number of terms, a technique called grouping can often be used. I. Order the terms from greatest to least degree. 2x 3 + 3x2 8x From the first two terms, factor out a GCF. From the second two terms, factor out a GCF. 3. Step 2 should produce two different groups that now share a GCF. Factor that GCF out of each group and write what remains from the two groups in a separate factor. 4. If either factor from step 3. can be further factored, then do so. I. Order the tenns from greatest to least degree. 2x 3 +x2-8x-9 2. From the first two terms, factor out a GCF. From the econd two terms, factor out a GCF. 3. Step 2 should produce two different groups that now share a GCF. Factor that GCF out of each group and write what remains from the two groups in a separate factor. 4. If either factor from step 3 can be further factored, then do so. Factor each of the following polynomials by grouping. x3 + 3x1-x-3 \ J ' ' X 'i( ')( -1-?>) - \ ( -+ ()(-It )( 2._ i) ( -\7 t C -+Dtx-\) 4- c -'-")-,( -4) ('I..- 4) ( 'k -i -,) ("-. - 4) ( \)(2. -
3 Factor each of the following polynomial functions whose equations and graphs are provided. Then, set each factor equal to zero and find the values of x. F(x) = 3x x2 + 12x 4 0 t=(-.c):: 3 ( 4x-+ 4) = 3>t (x -+2.)(x+2)...,... 1 ' +...,....jl....,....., , ,...,.... What is they- intercept of the graph? (0.. o) What is the constant tenn in the equation? Q G(x) = x 4-5.x2 + 4 {;..()t) ': (')( 2-_ 'i )( 'l-_ \ ) G,.("K-)-:... (X. )Q(-!>-)(X \)(_x. \) x.-\ 0 )( :..\,, -5 l \... L :...:...:...: What is they - intercept of the graph? ( 0, What is the constant tenn in the equation? 1 Hi(x) = 2x x2-2x - 5 \ ",I \.\C,.) : 'X. c to\- cs)-\ ( "T cs) : ( 2x-+ s)(x - l) - (2 9': s) C U(><. -I) +s o 2.)l-::.- x:.-%. )(-t\ 0 y..::. -\ -\:0 x, : : : r + + i : i i : 1 1 i ! + {.. 1 : ; : :.. What is they - intercept of the graph? What is the constant tem1 in the equation? -S-
4 P(x) = 2x 4-5.x2 + 3 ) 7- (i 2. -?>;(x -,) c :2. - )( +\ )(x-\) x.-?> :: = 3 Y:J1 ::! \. '11 5 x-t\: o =-\ r:::::::::::::r:. ::::::::r:::::::: ::::::::::::r::::::t:r:::::::::::::i! I i!......! "i" I !, - f : l i 1 L... L... L... wl..... What is they- intercept of the graph? (o' ) What is the constant tenn in the equation? ;.3,.
5 Factoring Practice Completely factor each of the following polynomials. Date Period x3-14.x2-6x 2. 6.x3-3x x x2-14.x3-6x 4 x( "x?- - 7x - 3) - 3x 5 -r tcx -s -td-1k -b)('-1- l )<.. 3 -Lf}(. -;i x: (_3x.-\- l ) ( --3) -3-,<()( L.\ -lx. 1 -Q) - ix :1 ( 3 x :1 -t 1 -+ i) -3x (x -\- )(')l,_ -'i-).2,.)( 2 ( 3x. -t \)( -t :l') l -3x (X:;-;))(x. ;-.).)(x -.;}.) 4. x3-3.x2-9x x2-2.x3 + 8x x3+18x x (x-?>)-,(x-3) ( -3) (x >- - Cf) (x-3)(x-t 3')(x-3) ( -+sxx - 3J 'J- - }. 3+3x :>.--+ 8x -1 -.:l. x ( x? - '1) -x?. '/..-3)+ - ) - cx+3)(x-3) ( -3)(- '1- :2. -t 4 J -{.:.. 3) (x. -\-)._) (x-;l..) 7. 2x 4 -.x x 4 -.x x x3-27x ( -;... 1 ;- 5)(_x i - 3) ('I.. :1. -t )(x i - 1 J 3. X ( 2$ 't -t?:,';... '2. - 9) {_y._')-+-3)( X X - :)_) 3x (2.)(. i_ 3)(x :).t- 3) Factor each of the po ynomial functions. Then, identify the zeros of the function. Show your work. 10.j{x)=5.x3-20x. 11. g(x)=3x 3-3.x2-18x 12. h(x)=-10.x3+26x2+l2x ) '5x.( i - 4-) 'jv-) == 3x( 2.. _ x - b) h6'):: -.l.x.( s 1 -\3)<..- ) ) sx(..x+i)(.x-c:lj )=- 3'AC:,..-3)()(.. \<1.) \,(2'.)= -d-x (s-- t- --3) sx o X-t:L=-D x- -=c 3x=-0 x-3:::.-o x t -.;l=o - x.. =o 5">,-+ -=-c ')( - 6-::.(:) X:::.. D X - x: = ')( ::.a )(:=-3 X ::.-,;t ::::. 0 X= -i 'X : 3
6 13. p(x) = x3 + 2x2-4x-8 {)()() :;. "'?. & -t :).)- L\ (x-t ). -=- (x+ (x 2 - \.\) ::. ( kj..)( -t-.?.-)(x-::l 14. q(x) = 3x3 + 5.x2-3x-5 Ci)'.:: )2{3x'*5)-,(3x+5 [j_) = (3X-+ s)(x 1 - ') ) =- G)(+S)C.X.-+\X',l r(x) = x4-10.x2 + 9 r&) C x?--'t)(x 'l.._ 1) r6<)::: l -ttq(-3)(_'t.--t\ - :t3 ::.o X. -3-:::-0 3x-\ 5::: D x.-+ \ :::.0 X-\-=-0 x - x -\ x=-\ x_::..-3 )(:::. 3,><..:::.-\ 16. m(x)=x 4-2x 3 -I5.x2 17. g(x)=4x3+16.x2+16x 18. h(x)=2x 3 +3.x2-8x-12 rr-1.j..) = :l e x )(_-\s frq0 ::: )'.. 1 ( x. -'5 )0,--I- 3) SJ-) = IJ-x.( x-.. ::>-)()..--\ d_:) y..,.')---=-0 X - 5 -::.0 \-3={: 1i-'l----:.-O X-t l. =0 X:=-:, X =--3 j(i() = '-t- (x 1 -t- Y-1' * v "<0 =?<,:l.('.l -t - Gx 3 )l..=-d f{,i)=- t-t{_ x?--ti) \-\(j<.) :=. (:2><. 3)( X.-+.).x_ x-, Pictured below is the graph of a polynomial function,.f{x). Use the graph to answer the questions that follow. 19. Identify the 20. Based on the graph, what are the coordinates zeros of.f{x). of they- intercept?!... )...!...! :...!... t '...! !...! : : : : : : : : : :...,... f! + l + 1 f l i H.. l... l ' Jil':" JtJJJ i_j_tvll- B? J LL! 22. Rewrite the factored equation in standard fonn. )'.:: )(.x-d-)(x-t,) W)::. ( x?--u/)[ X. +,) -f{ J-= 't. 3 + x.' L\ 21. Write the equation ofj{x) in factored form. 23. What is the constant term in the standard fonn equation of j{x)? State the connection that this term in the equation has with the graph of j{x). -\- (. 5 --'\.. IL.' II, d. - \..\ - _J - _-f..,.,. "- \. <...J v-=\- 1) > (OJ-4J-
2-2: Evaluate and Graph Polynomial Functions
2-2: Evaluate and Graph Polynomial Functions What is a polynomial? -A monomial or sum of monomials with whole number exponents. Degree of a polynomial: - The highest exponent of the polynomial How do we
More informationREVIEW, pages Chapter 1: Polynomial Expressions and Functions Review Solutions DO NOT COPY. P 1.1. Write the division statement.
REVIEW, pages 72 77 1.1 1. Use long division to divide 7x 3 + 6x 4-7x - 9x 2 + 8 by x 1. Write the division statement. Write the polynomial in descending order: 6x 4 7x 3 9x 2 7x 8 6x 4 6x 3 6x 3 13x 2
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
More informationChapter 4E - Combinations of Functions
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)?
More informationWarm Up answers. 1. x 2. 3x²y 8xy² 3. 7x² - 3x 4. 1 term 5. 2 terms 6. 3 terms
Warm Up answers 1. x 2. 3x²y 8xy² 3. 7x² - 3x 4. 1 term 5. 2 terms 6. 3 terms Warm Up Assignment 10/23/14 Section 6.1 Page 315: 2 12 (E) 40 58 (E) 66 Section 6.2 Page 323: 2 12 (E) 16 36 (E) 42 46 (E)
More informationTopic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3
Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring
More informationReview: complex numbers
October 5/6, 01.5 extra problems page 1 Review: complex numbers Number system The complex number system consists of a + bi where a and b are real numbers, with various arithmetic operations. The real numbers
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 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 informationUsing the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...-\; 0 or - l (- - ONLl --
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\ 4-8x 3-12x 2 + 24x {?\whose graph is
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 informationCan there be more than one correct factorization of a polynomial? There can be depending on the sign: -2x 3 + 4x 2 6x can factor to either
MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring
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 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 point-slope
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 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 informationTotal Possible Points = 150 Points. 1) David has 980 yards of fencing and wishes to enclose a rectangular area. (2.5 points) + '3 b. 7 + Ib+3, tf-.
MA180 Professor Fred Katiraie Test IT Form A (Fall 2007) Name: Total Possible Points = 150 Points 1) David has 980 yards of fencing and wishes to enclose a rectangular area. (2.5 points) a) Express the
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 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 informationTropical Polynomials
1 Tropical Arithmetic Tropical Polynomials Los Angeles Math Circle, May 15, 2016 Bryant Mathews, Azusa Pacific University In tropical arithmetic, we define new addition and multiplication operations on
More informationPolynomial Degree Leading Coefficient. Sign of Leading Coefficient
Chapter 1 PRE-TEST REVIEW Polynomial Functions MHF4U Jensen Section 1: 1.1 Power Functions 1) State the degree and the leading coefficient of each polynomial Polynomial Degree Leading Coefficient y = 2x
More informationSection 3.1: Characteristics of Polynomial Functions
Chapter 3: Polynomial Functions Section 3.1: Characteristics of Polynomial Functions pg 107 Polynomial Function: a function of the form f(x) = a n x n + a n 1 x n 1 +a n 2 x n 2 +...+a 2 x 2 +a 1 x+a 0
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationKEY CONCEPTS. Factoring is the opposite of expanding.
KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add
More informationMath 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
More informationpolynomial function polynomial function of degree n leading coefficient leading-term test quartic function turning point
polynomial function polynomial function of degree n leading coefficient leading-term test quartic function turning point quadratic form repeated zero multiplicity Graph Transformations of Monomial Functions
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 informationSections 7.1, 7.2: Sums, differences, products of polynomials CHAPTER 7: POLYNOMIALS
Sections 7.1, 7.2: Sums, differences, products of polynomials CHAPTER 7: POLYNOMIALS Quiz results Average 73%: high h score 100% Problems: Keeping track of negative signs x = + = + Function notation f(x)
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
More informationAlgebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials
Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +
More informationMathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017
Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More information2, or x 5, 3 x 0, x 2
Pre-AP Algebra 2 Lesson 2 End Behavior and Polynomial Inequalities Objectives: Students will be able to: use a number line model to sketch polynomials that have repeated roots. use a number line model
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 n-1,, a 2, a 1, a 0 be real
More informationFactors of Polynomials Factoring For Experts
Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product
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 informationSimplifying Rational Expressions and Functions
Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written
More informationPolynomials 6c Classifying the Zeros of a Polynomial Functions
Polynomials 6c Classifying the Zeros of a Polynomial Functions Standards: A APR.2, A APR.3, F IF.7c, N CN.9 Learning Target(s): How many zeros does a polynomial have? How can we find all the exact zeros
More informationQuarter 2 400, , , , , , ,000 50,000
Algebra 2 Quarter 2 Quadratic Functions Introduction to Polynomial Functions Hybrid Electric Vehicles Since 1999, there has been a growing trend in the sales of hybrid electric vehicles. These data show
More informationCh 7 Summary - POLYNOMIAL FUNCTIONS
Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)
More informationAlgebra 2, Chapter 5 Review
Name: Class: Date: Algebra 2, Chapter 5 Review 4.4.1: I can factor a quadratic expression using various methods and support my decision. 1. (1 point) x 2 + 14x + 48 2. (1 point) x 2 x + 42 3. (1 point)
More informationReview Notes - Solving Quadratic Equations
Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic
More informationIM 3 Assignment 4.3 : Graph Investigation Unit 4 Polynomial Functions
(A) Lesson Context BIG PICTURE of this UNIT: What is a Polynomial and how do they look? What are the attributes of a Polynomial? How do I work with Polynomials? CONTEXT of this LESSON: Where we ve been
More informationMath 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...
Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form... where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. 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 informationIM 3 Assignment 4.3 : Graph Investigation Unit 4 Polynomial Functions
(A) Lesson Context BIG PICTURE of this UNIT: What is a Polynomial and how do they look? What are the attributes of a Polynomial? How do I work with Polynomials? CONTEXT of this LESSON: Where we ve been
More informationSec 2.1 Operations with Polynomials Polynomial Classification and Operations
Sec.1 Operations with Polynomials Polynomial Classification and Operations Name Examples Non-Examples Monomial 1. x 4 degree:4 or quartic 1. x 4 (one term). a degree: or quadratic. 5 m. 5 degree:0 or constant.
More informationSection 6.5 A General Factoring Strategy
Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b
More informationSection 4.1: Polynomial Functions and Models
Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial
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 information8-1. Adding and Subtracting Polynomials. Warm- Up. Write in Standard Form. Write in Slope Intercept Form. 1.) y = -! x +2 2.
8-1 Adding and Subtracting Polynomials Warm- Up Write in Standard Form. Write in Slope Intercept Form. 1.) y = -! x +.) (3, 4) (6, 1)! Write an equation of the line that passes through the given point
More informationMath 0312 EXAM 2 Review Questions
Name Decide whether the ordered pair is a solution of the given system. 1. 4x + y = 2 2x + 4y = -20 ; (2, -6) Solve the system by graphing. 2. x - y = 6 x + y = 16 Solve the system by substitution. If
More informationReview for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.
LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in
More informationThe most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.
MOST FACTORED FORM The most factored form is the most factored version of a rational expression. Being able to find the most factored form is an essential skill when simplifying the derivatives found using
More informationChapter 3-1 Polynomials
Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents
More informationSection 3.1 Quadratic Functions
Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application
More informationSection 4.2 Polynomial Functions of Higher Degree
Section 4.2 Polynomial Functions of Higher Degree Polynomial Function P(x) P(x) = a degree 0 P(x) = ax +b (degree 1) Graph Horizontal line through (0,a) line with y intercept (0,b) and slope a P(x) = ax
More informationNeed help? Try or 4.1 Practice Problems
Day Date Assignment (Due the next class meeting) Friday 9/29/17 (A) Monday 10/9/17 (B) 4.1 Operations with polynomials Tuesday 10/10/17 (A) Wednesday 10/11/17 (B) 4.2 Factoring and solving completely Thursday
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 informationAdvanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial.
Advanced Math Quiz 3.1-3.2 Review Name: Dec. 2014 Use Synthetic Division to divide the first polynomial by the second polynomial. 1. 5x 3 + 6x 2 8 x + 1, x 5 1. Quotient: 2. x 5 10x 3 + 5 x 1, x + 4 2.
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 informationChapter REVIEW ANSWER KEY
TEXTBOOK HELP Pg. 313 Chapter 3.2-3.4 REVIEW ANSWER KEY 1. What qualifies a function as a polynomial? Powers = non-negative integers Polynomial functions of degree 2 or higher have graphs that are smooth
More informationPolynomial Functions
Polynomial Functions Equations and Graphs Characteristics The Factor Theorem The Remainder Theorem http://www.purplemath.com/modules/polyends5.htm 1 A cross-section of a honeycomb has a pattern with one
More informationHonors Advanced Mathematics November 4, /2.6 summary and extra problems page 1 Recap: complex numbers
November 4, 013.5/.6 summary and extra problems page 1 Recap: complex numbers Number system The complex number system consists of a + bi where a and b are real numbers, with various arithmetic operations.
More informationProblem Worth Score Total 14
MATH 241, Fall 14 Extra Credit Preparation for Final Name: INSTRUCTIONS: Write legibly. Indicate your answer clearly. Revise and clean up solutions. Do not cross anything out. Rewrite the page, I will
More informationAlgebra 32 Midterm Review Packet
Algebra 2 Midterm Review Packet Formula you will receive on the Midterm: x = b ± b2 4ac 2a Name: Teacher: Day/Period: Date of Midterm: 1 Functions: Vocabulary: o Domain (Input) & Range (Output) o Increasing
More informationMathB65 Ch 4 IV, V, VI.notebook. October 31, 2017
Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
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 informationUnit 7: Factoring Quadratic Polynomials
Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples
More informationl(- oo)-~j [-I <. )( L6\ \ -J ~ ~ ~~~ ~~L{ ,~:::-=r\ or L":: -j) {fevylemr.eor k, ("p J~ -4" e S ' e,~ :; ij or J iv I 0"'& ~~ a. 11 qa.
Algebra II Midterm Exam Review Solve: R view of Algebra 1 4. 215x + 31 = 16 /5xt3 1:: & 3 :: 2.1 3" ::. -J5 /,:::-=r\ or L":: -j) 2. II-2xl = 15 / j - ;).'1 -:.115 00( 1-).)(":.-15 - X-: 1"1 by:-t3-8 5
More informationLT1: Adding and Subtracting Polynomials. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms.
LT1: Adding and Subtracting Polynomials *When adding polynomials, simply combine like terms. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms. 1.
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More information(5) difference of squares,
EOCT REVIEW UNIT 5 Quadratic Functions Name Kut Write each expression in factored form. 1. X2-2x - 15 (X>5')(X f 3) 2. X2-18x + 81 (x:-q)(x-q) (1)' (X, ) z- Complete each square and write the resulting
More informationECEN 604: Channel Coding for Communications
ECEN 604: Channel Coding for Communications Lecture: Introduction to Cyclic Codes Henry D. Pfister Department of Electrical and Computer Engineering Texas A&M University ECEN 604: Channel Coding for Communications
More informationPolynomial Functions. x n 2 a n. x n a 1. f x = a o. x n 1 a 2. x 0, , a 1
Polynomial Functions A polynomial function is a sum of multiples of an independent variable raised to various integer powers. The general form of a polynomial function is f x = a o x n a 1 x n 1 a 2 x
More informationQuick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors.
Lesson 11-3 Quick-and-Easy Factoring BIG IDEA Some polynomials can be factored into polynomials of lower degree; several processes are available to fi nd factors. Vocabulary factoring a polynomial factored
More informationPolynomial Review Problems
Polynomial Review Problems 1. Find polynomial function formulas that could fit each of these graphs. Remember that you will need to determine the value of the leading coefficient. The point (0,-3) is on
More informationPolynomials Patterns Task
Polynomials Patterns Task Mathematical Goals Roughly sketch the graphs of simple polynomial functions by hand Graph polynomial functions using technology Identify key features of the graphs of polynomial
More informationThe degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =
Math 1310 A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function is n. We call the term the leading term,
More informationInverse Variation. y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0.
Inverse Variation y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0. Inverse variation xy = k or y = k where k is not equal to 0. x Identify whether the following functions
More informationGraphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).
Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a
More informationSolutions to Exercises, Section 2.5
Instructor s Solutions Manual, Section 2.5 Exercise 1 Solutions to Exercises, Section 2.5 For Exercises 1 4, write the domain of the given function r as a union of intervals. 1. r(x) 5x3 12x 2 + 13 x 2
More informationUnit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to
Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to Learning Target #1: Factoring Factor the GCF out of a polynomial Factor a polynomial when a = 1 Factor
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationRewriting Absolute Value Functions as Piece-wise Defined Functions
Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
More informationMAC 1147 Final Exam Review
MAC 1147 Final Exam Review nstructions: The final exam will consist of 15 questions plu::; a bonus problem. Some questions will have multiple parts and others will not. Some questions will be multiple
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 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 information2(x 4 7x 2 18) 2(x 2 9)(x 2 + 2) 2(x 3)(x + 3)(x 2 + 2)
Completely factor 2x 4 14x 2 36 2(x 4 7x 2 18) 2(x 2 9)(x 2 + 2) 2(x 3)(x + 3)(x 2 + 2) Add and simplify Simplify as much as possible Subtract and simplify Determine the inverse of Multiply and simplify
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
6-7 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 informationPRINTABLE VERSION. Practice Final. Question 1. Find the coordinates of the y-intercept for 5x 9y + 6 = 0. 2 (0, ) 3 3 (0, ) 2 2 (0, ) 3 6 (0, ) 5
PRINTABLE VERSION Practice Final Question Find the coordinates of the y-intercept for 5x 9y + 6 = 0. (0, ) (0, ) (0, ) 6 (0, ) 5 6 (0, ) 5 Question Find the slope of the line: 7x 4y = 0 7 4 4 4 7 7 4 4
More information1.1 : (The Slope of a straight Line)
1.1 : (The Slope of a straight Line) Equations of Nonvertical Lines: A nonvertical line L has an equation of the form y mx b. The number m is called the slope of L and the point (0, b) is called the y-intercept.
More informationSection IV.23. Factorizations of Polynomials over a Field
IV.23 Factorizations of Polynomials 1 Section IV.23. Factorizations of Polynomials over a Field Note. Our experience with classical algebra tells us that finding the zeros of a polynomial is equivalent
More informationCharacteristics of Polynomials and their Graphs
Odd Degree Even Unit 5 Higher Order Polynomials Name: Polynomial Vocabulary: Polynomial Characteristics of Polynomials and their Graphs of the polynomial - highest power, determines the total number of
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationPolynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions
CCGPS Advanced Algebra Polynomial Functions Polynomial Functions Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the
More informationMA094 Part 2 - Beginning Algebra Summary
MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationGraphs of Polynomial Functions
Graphs of Polynomial Functions By: OpenStaxCollege The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link]. Year 2006 2007 2008 2009 2010 2011 2012 2013
More informationComplete the following table using the equation and graphs given:
L2 1.2 Characteristics of Polynomial Functions Lesson MHF4U Jensen In section 1.1 we looked at power functions, which are single-term polynomial functions. Many polynomial functions are made up of two
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More informationSection 3.2 Polynomial Functions and Their Graphs
Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P (x) = 3, Q(x) = 4x 7, R(x) = x 2 + x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 + 2x +
More information