HS Algebra II Semester 1 Module 1: Polynomial, Rational .
HIGLEY UNIFIED SCHOOL DISTRICTINSTRUCTIONAL ALIGNMENTHS Algebra II Semester 1Module 1: Polynomial, Rational, and Radical Relationships (45 days)Topic A: Polynomials β From Base Ten to Base X (11 instructional days)In Topic A, students draw on their foundation of the analogies between polynomial arithmetic and base ten computation, focusing on properties of operations, particularly the distributive property. InLesson 1, students write polynomial expressions for sequences by examining successive differences. They are engaged in a lively lesson that emphasizes thinking and reasoning about numbers andpatterns and equations. In Lesson 2, they use a variation of the area model referred to as the tabular method to represent polynomial multiplication and connect that method back to application of thedistributive property.In Lesson 3, students continue using the tabular method and analogies to the system of integers to explore division of polynomials as a missing factor problem. In this lesson, students also take time toreflect on and arrive at generalizations for questions such as how to predict the degree of the resulting sum when adding two polynomials. In Lesson 4, students are ready to ask and answer whetherlong division can work with polynomials too and how it compares with the tabular method of finding the missing factor. Lesson 5 gives students additional practice on all operations with polynomialsand offers an opportunity to examine the structure of expressions such as recognizing that π(π 1)(2π 1) /6is a 3rd degree polynomial expression with leading coefficient 13 without having to expand it out.In Lesson 6, students extend their facility with dividing polynomials by exploring a more generic case; rather than dividing by a factor such as (π₯ 3), they divide by the factor (π₯ π) or (π₯ π). This gives22them the opportunity to discover the structure of special products such as (π₯ π)(π₯ ππ₯ π ) in Lesson 7 and go on to use those products in Lessons 8β10 to employ the power of algebra over thecalculator. In Lesson 8, they find they can use special products to uncover mental math strategies and answer questions such as whether or not 2100 1 is prime. In Lesson 9, they consider how theseproperties apply to expressions that contain square roots. Then, in Lesson 10, they use special products to find Pythagorean triples.The topic culminates with Lesson 11 and the recognition of the benefits of factoring and the special role of zero as a means for solving polynomial equations.Big Idea: Polynomials form a system analogous to the integers.Polynomials can generalize the structure of our place value system and of radical expressions.EssentialQuestions: How is polynomial arithmetic similar to integer arithmetic?What does the degree of a polynomial tell you about its related polynomial rical symbol, variable symbol, algebraic expression, numerical expression, monomial, binomial, polynomial expression, sequence, arithmetic sequence,equivalent polynomial expressions, polynomial identity, coefficient of a monomial, terms of a polynomial, like terms of a polynomial, standard form of apolynomial in one variable, degree of a polynomial in one variable, conjugate, Pythagorean Theorem, converse to the Pythagorean Theorem, PythagoreanTriple, polynomial function, degree of a polynomial function, constant function, linear function, quadratic function, cubic function, zeros or roots of a functionGalileo: Geometry Module 1 Foundational Skills Assessment; Galileo: Topic A AssessmentCommon Core StandardsExplanations & ExamplesResourcesPage 1 of 31
A. Interpret the structure of expressionsUse the structure of an expression to identify ways to442 22 2rewrite it. For example, see x β y as (x ) β (y ) , thusrecognizing it as a difference of squares that can be2222factored as (x β y )(x y ).Note: If students have trouble with evaluating or simplifyingexpressions or solving equations, then you might want to revisitLessons 6β9 in Grade 9, Module 1, and Lesson 2 in Grade 9, Module 4.Eureka Math:Module 1 Lesson 2 - 9Module 1 Lesson 10-11A.SSE.A.2Eureka Math:Module 1 Lesson 2 β 7Module 1 Lesson 10C. Use polynomial identities to solve problemsA.APR.C.4MP.1Prove polynomial identities and use them to describenumerical relationships. For example, the polynomial2 222 22identity (x y )2 (x β y ) (2xy) can be used togenerate Pythagorean triples.Make sense of problems and persevere in solvingthem.Students discover the value of equating factored terms of a polynomialto zero as a means of solving equations involving polynomials.Eureka Math:Module 1 Lesson 1Module 1 Lesson 2Module 1 Lesson 11Reason abstractly and quantitatively.Students apply polynomial identities to detect prime numbers anddiscover Pythagorean triples.Eureka Math:Module 1 Lesson 4Module 1 Lesson 8MP.212/17/2014Page 2 of 31
Construct viable arguments and critique thereasoning of others.MP.3Attend to precision.MP.6Look for and make use of structure.MP.7Look for and express regularity in repeatedreasoning.MP.812/17/2014Mathematically proficient students understand and use statedassumptions, definitions, and previously established results inconstructing arguments. They make conjectures and build a logicalprogression of statements to explore the truth of their conjectures.They are able to analyze situations by breaking them into cases, andcan recognize and use counterexamples. They justify their conclusions,communicate them to others, and respond to the arguments of others.Mathematically proficient students try to communicate precisely toothers. They try to use clear definitions in discussion with others and intheir own reasoning. They state the meaning of the symbols theychoose, including using the equal sign consistently and appropriately.They calculate accurately and efficiently, express numerical answerswith a degree of precision appropriate for the problem context.Students connect long division of polynomials with the long-divisionalgorithm of arithmetic and perform polynomial division in an abstractsetting to derive the standard polynomial identities.Eureka Math:Module 1 Lesson 5Module 1 Lesson 7Module 1 Lesson 8, 9Students understand that polynomials form a system analogous to theintegers. Students apply polynomial identities to detect prime numbersand discover Pythagorean triples. Students recognize factors ofexpressions and develop factoring techniques.Eureka Math:Module 1 Lesson 1 - 4Module 1 Lesson 6Module 1 Lesson 8Module 1 LessonModule 1 LessonEureka Math:Module 1 Lesson 10Eureka Math:Module 1 Lesson 1 β 6Module 1 Lesson 8, 9, 10Page 3 of 31
HS Algebra II Semester 1Module 1: Polynomial, Rational, and Radical Relationships (45 days)Topic B: Factoring β Its Use and Its Obstacles (10 instructional days)Armed with a newfound knowledge of the value of factoring, students develop their facility with factoring and then apply the benefits to graphing polynomial equations in Topic B. InLessons 12β13, students are presented with the first obstacle to solving equations successfully. While dividing a polynomial by a given factor to find a missing factor is easilyaccessible, factoring without knowing one of the factors is challenging. Students recall the work with factoring done in Algebra I and expand on it to master factoring polynomialswith degree greater than two, emphasizing the technique of factoring by grouping.In Lessons 14β15, students find that another advantage to rewriting polynomial expressions in factored form is how easily a polynomial function written in this form can be graphed.Students read word problems to answer polynomial questions by examining key features of their graphs. They notice the relationship between the number of times a factor isrepeated and the behavior of the graph at that zero (i.e., when a factor is repeated an even number of times, the graph of the polynomial will touch the π₯-axis and βbounceβ backoff, whereas when a factor occurs only once or an odd number of times, the graph of the polynomial at that zero will βcut throughβ the π₯-axis). In these lessons, students willcompare hand plots to graphing- calculator plots and zoom in on the graph to examine its features more closely.In Lessons 16β17, students encounter a series of more serious modeling questions associated with polynomials, developing their fluency in translating between verbal, numeric,algebraic, and graphical thinking. One example of the modeling questions posed in this lesson is how to find the maximum possible volume of a box created from a flat piece ofcardboard with fixed dimensions.In Lessons 18β19, students are presented with their second obstacle: βWhat if there is a remainder?β They learn the Remainder Theorem and apply it to further understand theconnection between the factors and zeros of a polynomial and how this relates to the graph of a polynomial function. Students explore how to determine the smallest possibledegree for a depicted polynomial and how information such as the value of the π¦-intercept will be reflected in the equation of the polynomial.The topic culminates with two modeling lessons (Lessons 20β21) involving approximating the area of the cross-section of a riverbed to model the volume of flow. The problemdescription includes a graph of a polynomial equation that could be used to model the situation, and students are challenged to find the polynomial equation itself. Big Idea: EssentialQuestions:VocabularyAssessmentsWhat impact does an even- or odd-degree polynomial function have on its graph?How do polynomials helps solve real-world problems?Difference of squares identity, multiplicities, zeros or roots, relative maximum (maxima), relative minimum (minima), end behavior, even function, oddfunction, remainder theorem, factor theoremGalileo: Topic B AssessmentStandard12/17/2014Common Core StandardsExplanations & ExamplesResourcesPage 4 of 31
N.Q.A.2A. Reason qualitatively and units to solveproblemsDefine appropriate quantities for the purpose ofdescriptive modeling.A.SSE.A.2A. Interpret the structure of expressionsUse the structure of an expression to identify ways to442 22 2rewrite it. For example, see x β y as (x ) β (y ) , thusrecognizing it as a difference of squares that can be2222factored as (x β y )(x y ).12/17/2014Eureka Math:Module 1 Lesson 15, 16,17, 20, 21Eureka Math:Module 1 Lesson 12 β 14Module 1 Lesson 17 - 21Page 5 of 31
A.APR.B.2Eureka Math:Module 1 Lesson 19 - 21B. Understand the relationship between zerosand factors of polynomialsKnow and apply the Remainder Theorem: For apolynomial p(x) and a number a, the remainder ondivision by x β a is p(a), so p(a) 0 if and only if (x β a) isa factor of p(x).The Remainder theorem says that if a polynomial p(x) is divided by x βa, then the remainder is the constant p(a). That is,p(x ) q(x)(x a) p(a). So if p(a) 0 then p(x) q(x)(x-a).Include problems that involve interpreting the Remainder Theoremfrom graphs and in problems that require long division.A.APR.B.3B. Understand the relationship between zerosand factors of polynomials12/17/2014Eureka Math:Module 1 Lesson 14, 15,Page 6 of 31
Identify zeros of polynomials when suitablefactorizations are available, and use the zeros toconstruct a rough graph of the function defined by thepolynomial.A.APR.D.6D. Rewrite rational expressionsRewrite simple rational expressions in different forms;write a(x)/b(x) in the form q(x) r(x)/b(x), where a(x),b(x), q(x), and r(x) are polynomials with the degree ofr(x) less than the degree of b(x), using inspection, longdivision, or, for the more complicated examples, acomputer algebra system.F.IF.C.7cC. Analyze functions using differentrepresentationGraph functions expressed symbolically and show keyfeatures of the graph, by hand in simple cases and usingtechnology for more complicated cases.17, 19, 20, 21The polynomial q(x) is called the quotient and the polynomial r(x) iscalled the remainder. Expressing a rational expression in this formallows one to see different properties of the graph, such as horizontalasymptotes.Eureka Math:Module 1 Lesson 12, 13,18, 19, 20, 21Eureka Math:Module 1 Lesson 14, 15,17, 19, 20, 21c. Graph polynomial functions, identifying zeros whensuitable factorizations are available, and showing endbehavior.12/17/2014Page 7 of 31
MP.1Make sense of problems and persevere in solvingthem.Students discover the value of equating factored terms of a polynomialto zero as a means of solving equations involving polynomials.MP.2Reason abstractly and quantitatively.MP.3Construct viable arguments and critique thereasoning of others.MP.5Use appropriate tools strategically.MP.7Look for and make use of structure.Students apply polynomial identities to detect prime numbers anddiscover Pythagorean triples. Students also learn to make sense ofremainders in polynomial long division problems.Mathematically proficient students understand and use statedassumptions, definitions, and previously established results inconstructing arguments. They make conjectures and build a logicalprogression of statements to explore the truth of their conjectures.They are able to analyze situations by breaking them into cases, andcan recognize and use counterexamples. They justify their conclusions,communicate them to others, and respond to the arguments of others.They reason inductively about data, making plausible arguments thattake into account the context from which the data arose.Mathematically proficient students are also able to compare theeffectiveness of two plausible arguments, distinguish correct logic orreasoning from that which is flawed, andβif there is a flaw in anargumentβexplain what it is.Mathematically proficient students consider the available tools whensolving a mathematical problem. These tools might include pencil andpaper, concrete models, a ruler, a protractor, a calculator, aspreadsheet, a computer algebra system, a statistical package, ordynamic geometry software. Proficient students are sufficientlyfamiliar with tools appropriate for their grade or course to make sounddecisions about when each of these tools might be helpful, recognizingboth the insight to be gained and their limitations. For example,mathematically proficient high school students analyze graphs offunctions and solutions generated using a graphing calculator. Theydetect possible errors by strategically using estimation and othermathematical knowledge. When making mathematical models, theyknow that technology can enable them to visualize the results ofvarying assumptions, explore consequences, and compare predictionswith data. They are able to use technological tools to explore anddeepen their understanding of concepts.Students connect long division of polynomials with the long-divisionalgorithm of arithmetic and perform polynomial division in an abstractsetting to derive the standard polynomial identities. Studentsrecognize structure in the graphs of polynomials in factored form anddevelop refined techniques for graphing.12/17/2014Eureka Math:Module 1 Lesson 12Module 1 Lesson 20Eureka Math:Module 1 Lesson 17Eureka Math:Module 1 Lesson 14, 15,16, 17Eureka Math:Module 1 Lesson 14Module 1 Lesson 21Eureka Math:Module 1 Lesson 12Module 1 Lesson 13Module 1 Lesson 14Module 1 Lesson 18Page 8 of 31
Module 1 Lesson 20MP.8Look for and express regularity in repeatedreasoning.12/17/2014Students understand that polynomials form a system analogous to theintegers. Students apply polynomial identities to detect primenumbers and discover Pythagorean triples. Students recognize factorsof expressions and develop factoring techniques.Eureka Math:Module 1 Lesson 15Module 1 Lesson 19Page 9 of 31
HS Algebra II Semester 1Module 1: Polynomial, Rational, and Radical Relationships (45 days)Topic C: Solving and Applying Equations β Polynomial, Rational and Radical (14 instructional days)In Topic C, students continue to build upon the reasoning used to solve equations and their fluency in factoring polynomial expressions. In Lesson 22, students expand theirunderstanding of the division of polynomial expressions to rewriting simple rational expressions (A-APR.D.6) in equivalent forms. In Lesson 23, students learn techniques forcomparing rational expressions numerically, graphically, and algebraically. In Lessons 24β25, students learn to rewrite simple rational expressions by multiplying, dividing, adding, orsubtracting two or more expressions. They begin to connect operations with rational numbers to operations on rational expressions. The practice of rewriting rational expressions inequivalent forms in Lessons 22β25 is carried over to solving rational equations in Lessons 26 and 27. Lesson 27 also includes working with word problems that require the use ofrational equations. In Lessons 28β29, we turn to radical equations. Students learn to look for extraneous solutions to these equations as they did for rational equations.In Lessons 30β32, students solve and graph systems of equations including systems of one linear equation and one quadratic equation and systems of two quadratic equations. Next,in Lessons 33β35, students study the definition of a parabola as they first learn to derive the equation of a parabola given a focus and a directrix and later to create the equation ofthe parabola in vertex form from the coordinates of the vertex and the location of either the focus or directrix. Students build upon their understanding of rotations and translations2from Geometry as they learn that any given parabola is congruent to the one given by the equation π¦ ππ₯ for some value of π and that all parabolas are similar. Systems of non-linear functions create solutions more complex than those of systems of linear functions. Mathematicians use the focus and directix of a parabola to derive an equation.Big Idea:EssentialQuestions:VocabularyAssessments How do you reduce a rational expression to lowest terms? How do you compare the values of rational expressions? Why is it important to check the solutions of a rational or radical equation? Why are solving systems of nonlinear functions different than systems of linear functions? What does the focus and directix define a parabola? What conditions will two parabolas be congruent?Rational expression, complex fraction, equating numerators method, equating fractions method, extraneous solution, linear systems, parabola, axis ofsymmetry of a parabola, vertex of a parabola, paraboloid, focus, directrix, conic sections, eccentricity, vertical scaling, horizontal scaling, dilationGalileo: Topic C AssessmentStandardA.APR.D.6Common Core StandardsD. Rewrite rational expressionsRewrite simple rational expressions in different forms;write a(x)/b(x) in the form q(x) r(x)/b(x), where a(x),12/17/2014Explanations & ExamplesResourcesEureka Math:Module 1 Lesson 22 - 27Page 10 of 31
b(x), q(x), and r(x) are polynomials with the degree ofr(x) less than the degree of b(x), using inspection, longdivision, or, for the more complicated examples, acomputer algebra system. A.APR.D.7D. Rewrite rational expressionsUnderstand that rational expressions form a systemanalogous to the rational numbers, closed underaddition, subtraction, multiplication, and division by anonzero rational expression; add, subtract, multiply,and divide rational expressions.A.REI.A.1A. Understand solving equations as a process ofreasoning and explain the reasoningExplain each step in solving a simple equation asfollowing from the equality of numbers asserted at theprevious step, starting from the assumption that theoriginal equation has a solution. Construct a viableargument to justify a solution method.A.REI.A.2A. Understand solving equations as a process ofreasoning and explain the reasoningHONORS ONLYA major theme of the module is AβAPR.7. Teachers should continually remindstudents of the connections between rational expressions and rationalnumbers as students add, subtract, multiply and divide rational expressions.Examples: Use the formula for the sum of two fractions to explain whythe sum of two rational expressions is another rationalexpression. Expressin the form π π₯b(x) are polynomials.π₯ , where a(x) andIn Algebra II, tasks are limited to simple rational or radical equations.Properties of operations can be used to change expressions on eitherside of the equation to equivalent expressions. In addition, adding thesame term to both sides of an equation or multiplying both sides by anon-zero constant produces an equation with the same solutions.Other operations, such as squaring both sides, may produce equationsthat have extraneous solutions.Examples: Explain why the equation x/2 7/3 5 has the same solutionsas the equation 3x 14 30. Does this mean that x/2 7/3 isequal to 3x 14? Show that x 2 and x -3 are solutions to the equationπ₯π₯Write the equation in a form that shows these arethe only solutions, explaining each step in your reasoning.Examples: Eureka Math:Module 1 Lesson 24-25Module 1 Lesson 28-29Eureka Math:Module 1 Lesson 24-29Solve simple rational and radical equations in onevariable, and give examples showing how extraneoussolutions may arise.12/17/2014Page 11 of 31
A.REI.B.4bB. Solve equations and inequalities in onevariableSolve quadratic equations in one variable.b. Solve quadratic equations by inspection (e.g., forx2 49), taking square roots, completing the square,the quadratic formula and factoring, as appropriate tothe initial form of the equation. Recognize when thequadratic formula gives complex solutions and writethem as a bi for real numbers a and b.A.REI.C.6C. Solve systems of equationsIn Algebra II, in the case of equations having roots with nonzeroimaginary parts, students write the solutions as a /-bi where a and bare real numbers.Eureka Math:Module 1 Lesson 31In Algebra II, tasks are limited to 3x3 systems.Eureka Math:Module 1 Lesson 30-31Example:Eureka Math:Module 1 Lesson 31Solve systems of linear equations exactly andapproximately (e.g., with graphs), focusing on pairs oflinear equations in two variables.A.REI.C.7C. Solve systems of equationsSolve a simple system consisting of a linear equationand a quadratic equation in two variables algebraicallyand graphically. For example, find the points ofintersection between the line22y β3x and the circle x y 3.12/17/2014Page 12 of 31
G.GPE.A.2Eureka Math:Module 1 Lesson 33-34A. Translate between the geometric descriptionand the equation for a conic sectionDerive the equation of a parabola given a focus and directrix.Given a focus and a directrix, create an equation for a parabola.Focus: π (π,π)Directrix: π-axisParabola: π· {(π,π) (π,π) is equidistant to F and to the π-axis.}Let π¨ be any point (π,) on the parabola π·. Let πβ² be a point on thedirectrix with the same π-coordinate as point π¨.What is the length of π¨Fβ²? π¨πβ² πUse the distance formula to create an expression that represents thelength of AF. π¨F sqtrt(π π)π (π π)πIdentify the focus and directrix of the parabola given by y2 -4xIdentify the focus and directrix of the parabola given by x2 12yWrite the standard form of the equation of the parabola with itsVertex at (0,0) and focus at (0, β4)Write the standard form of the equation of the parabola with itsvertex at (0, 0) and directrix y 5Write the standard form of the equation of the parabola with itsvertex at (0, 0) and directrix x 212/17/2014Page 13 of 31
MP.1Make sense of problems and persevere in solvingthem.Students solve systems of linear equations and linear and quadraticpairs in two variables. Further, students come to understand that the2complex number system provides solutions to the equation x 1 0and higher-degree equations.Eureka Math:Module 1 Lesson 26-27Module 1 Lesson 29-31Module 1 Lesson 33MP.2Reason abstractly and quantitatively.Students apply polynomial identities to detect prime numbers anddiscover Pythagorean triples. Students also learn to make sense ofremainders in polynomial long division problems.Eureka Math:Module 1 Lesson 23Module 1 Lesson 27Module 1 Lesson 34MP.3Construct viable arguments and critique thereasoning of others.Mathematically proficient students understand and use statedassumptions, definitions, and previously established results inconstructing arguments. They make conjectures and build a logicalprogression of statements to explore the truth of their conjectures.They are able to analyze situations by breaking them into cases, andcan recognize and use counterexamples. They justify their conclusions,communicate them to others, and respond to the arguments of others.12/17/2014Eureka Math:Module 1 Lesson 23Module 1 Lesson 28-29Module 1 Lesson 34-35Page 14 of 31
MP.4Model with mathematics.Students use primes to model encryption. Students transition betweenverbal, numerical, algebraic, and graphical thinking in analyzing appliedpolynomial problems. Students model a cross-section of a riverbedwith a polynomial, estimate fluid flow with their algebraic model, andfit polynomials to data. Students model the locus of points at equaldistance between a point (focus) and a line (directrix) discovering theparabola.Eureka Math:Module 1 Lesson 27Module 1 Lesson 33MP.5Use appropriate tools strategically.Mathematically proficient students consider the available tools whensolving a mathematical problem. These tools might include pencil andpaper, concrete models, a ruler, a protractor, a calculator, aspreadsheet, a computer algebra system, a statistical package, ordynamic geometry software. Proficient students are sufficientlyfamiliar with tools appropriate for their grade or course to make sounddecisions about when each of these tools might be helpful, recognizingboth the insight to be gained and their limitations. For example,mathematically proficient high school students analyze graphs offunctions and solutions generated using a graphing calculator. Theydetect possible errors by strategically using estimation and othermathematical knowledge.Eureka Math:Module 1 Lesson 31MP.7Look for and make use of structure.Students connect long division of polynomials with the long-divisionalgorithm of arithmetic and perform polynomial division in an abstractsetting to derive the standard polynomial identities. Students recognizestructure in the graphs of polynomials in factored form and developrefined techniques for graphing. Students discern the structure ofrational expressions by comparing to analogous arithmetic problems.Students perform geometric operations on parabolas to discovercongruence and similarity.Eureka Math:Module 1 Lesson 22Module 1 Lesson 24-26Students understand that polynomials form a system analogous to theintegers. Students apply polynomial identities to detect prime numbersand discover Pythagorean triples. Students recognize factors ofexpressions and develop factoring techniques. Further, studentsunderstand that all quadratics can be written as a product of linearfactors in the complex realm.Eureka Math:Module 1 Lesson 22Module 1 Lesson 31MP.8Look for and express regularity in repeatedreasoning.12/17/2014Module 1 Lesson 28-30Module 1 Lesson 34Page 15 of 31
HS Algebra II Semester 1Module 1: Polynomial, Rational, and Radical Relationships (45 days)Topic D: A Surprise from Geometry-Complex Numbers Overcome All Obstacles (5 instructional days)In Topic D, students extend their facility with finding zeros of polynomials to include complex zeros. Lesson 36 presents a third obstacle to using factors of polynomials to solve polynomial equations.Students begin by solving systems of linear and non-linear equations to which no real solutions exist, and then relate this to the possibility of quadratic equations with no real solutions. Lesson 37introduces complex numbers through their relationship to geometric transformations. That is, students observe that scaling all numbers on a number line by a factor of 1 turns the number line out ofits one-dimensionality and rotates it 180 through the plane. They then answer the question, βWhat scale factor could be used to create a rotation of 90 ?β In Lesson 38, students discover that complexnumbers have real uses; in fact, they can be used in finding real solutions of polynomial equations. In Lesson 39, students develop facility with properties and operations of complex numbers and thenapply that facility to factor polynomials with complex zeros. Lesson 40 brings the module to a close with the result that every polynomial can be rewritten as the product of linear factors, which is notpossible without complex numbers. Even though standards N-CN.C.8 and N-CN.C.9 are not assessed at the Algebra II level, they are included instructionally to develop further conceptual understanding. Every polynomial can be rewritten as the product of linear factors.Big Idea: EssentialQuestions:VocabularyAssessmentsComplex numbers, imaginary, discriminant, conjugate pairs, [Fundamental Theorem of Algebra (Honors only)]Galileo: Topic D AssessmentStandardN.CN.A.1Common Core StandardsExplanations & ExamplesA. Perform arithmetic operations withcomplex numbers2Know there is a complex number i such that i 1, andevery complex number has the form a bi with a and breal.12/17/2014Multiplying by I rotates every complex number in the complex plane by ππ about theorigin.Every complex number is in the form π πi, where π is the real part and π is theimaginary part of the number. Real numbers are also complex numbers; the real numberπ can be written as the complex number π πi.CommentsEureka Math:Module 1 Lesson 37Page 16 of 31
N.CN.A.2A. Perform arithmetic operations withcomplex numbers2Use the relation i β1 and the commutative,associative, and distributive properties to add, subtract,and multiply complex numbers.12/17/2014Addition and subtraction with complex numbers:(π i) (π πi ) (π π) ( π)iMultiplication with complex numbers2(π i) (π πi ) πc ci πdi d π (πc d ) ( c πd)iEureka Math
a factor of p(x). The Remainder theorem says that if a polynomial p(x) is divided by x β a, then the remainder is the constant p(a). That is, p(x ) q(x)(x So a if p(a) 0 then p(x) q(x)(x) p(a).-a). Include problems that involve interpreting the Remainder Theorem from graphs
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