Download Finding Real and Complex Zeros of Polynomials

Precalculus, Quarter 1, Unit 1.2
Finding Real and Complex Zeros of Polynomials
Overview
Number of instruction days:
7–9
(1 day = 53 minutes)
Content to Be Learned
Mathematical Practices to Be Integrated

Identify zeros of polynomials, both real and
complex.
5 Use appropriate tools strategically.

Apply the Binomial Theorem for the expansion
of (x + y)n and explore the relationship between
it and Pascal’s Triangle.

Use the Remainder Theorem to determine
zeros of polynomial functions.

Perform operations on complex numbers by
using commutative, associative, and
distributive properties.

Use graphing calculators to find zeros of
polynomial functions when suitable
factorizations are not available.

Use graph paper to sketch higher degreed
polynomials using characteristics.
7 Look for and make use of structures.

Find a pattern using the Binomial Theorem and
relate it to Pascal’s Triangle.
8 Look for and express regularity and repeated
reasoning.

Recognize the result of multiplying a complex
number by its conjugate: (a + bi)(a – bi) =
a2 + b2.
Essential Questions

What is the relationship between complex
numbers and real numbers?

How are the Binomial Theorem and Pascal’s
Triangle related?

What are the various methods for finding zeros
of a function?

How do the zeros of a polynomial help us in
constructing its graph?

Why are zeros important in our study of
polynomials?

Why is it important to know how to find the
conjugate of a complex number?
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Precalculus, Quarter 1, Unit 1.2
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Finding Real and Complex Zeros of Polynomials (7–9 days)
Standards
Common Core State Standards for Mathematical Content
Number and Quantity
The Complex Number System
N-CN
Perform arithmetic operations with complex numbers.
Use the relation i2 = –1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
N-CN.2
Algebra
Arithmetic with Polynomials and Rational Expressions
A-APR
Understand the relationship between zeros and factors of polynomials
A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
Use polynomial identities to solve problems
A-APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y
for a positive integer n, where x and y are any numbers, with coefficients determined for
example by Pascal’s Triangle.1
1
The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.
ACT’s College Readiness Standards: Mathematics
E.2.d
Find the rational roots, real roots, and complex roots of a polynomial function.
E.1.b
Use technology to approximate the real roots of a polynomial equation.
E.2.e
Describe the binomial theorem and Pascal’s triangle; use them to expand polynomials.
Common Core State Standards for Mathematical Practice
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
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Finding Real and Complex Zeros of Polynomials (7–9 days)
Precalculus, Quarter 1, Unit 1.2
Version 4
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
8
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them
to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically
proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In Algebra I, students learned to add, subtract, and multiply polynomials that simplify to forms that are
linear or quadratic. In Algebra II, students performed all arithmetic operation on polynomials, including
polynomials beyond the quadratics. They also began the study of the relationship between zeros and
factors of polynomials. They used the Remainder Theorem, the Binomial Theorem, and complex
numbers.
Current Learning
Students perform arithmetic operations with polynomials, including synthetic and long division. They
apply the Remainder Theorem and identify the zeros when suitable factorizations are available. They also
use zeros to construct a rough graph of the polynomial. Students learn that zeros can be found by hand
when factorization is suitable and that in more complex cases technology should be used. They expand
the Binomial Theorem and use it in the expansion of polynomial coefficients. They relate similarities and
differences between the Binomial Theorem and Pascal’s Triangle.
Providence Public Schools
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Precalculus, Quarter 1, Unit 1.2
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Finding Real and Complex Zeros of Polynomials (7–9 days)
Future Learning
In AP Calculus AB, students will need to have mastered solving any type of equation. In AP Calculus
BC, students will learn Euler’s Method of approximating zeros of a function. In AP Statistics, students
will use the Binomial Model, which is derived from the Binomial Theorem, for computing probabilities.
Additional Findings
There are no additional findings for this unit.
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.

Understand the relationship between zeros and factors of polynomials.

Use various methods to find real and complex zeros of polynomial functions.

Use zeros to construct rough graphs of polynomial functions.

Apply the Binomial Theorem and Pascal’s Triangle to expand binomials.

Simplify complex numbers in rectangular form.
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Providence Public Schools
Finding Real and Complex Zeros of Polynomials (7–9 days)
Precalculus, Quarter 1, Unit 1.2
Version 4
Instruction
Learning Objectives
Students will be able to:

Determine real and complex roots of polynomial equations.

Solve quadratic equations.

Use the Remainder Theorem and a variety of techniques (including factoring) to solve for zeros of
polynomials.

Approximate real zeros of polynomial functions.

Sketch a graph of a polynomial function, given the zeros.

Add, subtract, multiply, and divide complex numbers in rectangular form.

Use the Binomial Theorem to expand binomials using Pascal’s triangle.

Demonstrate understanding of concepts and skills related real and complex zeros of polynomials.
Resources
Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition
and Student Edition

Sections 4-1 – 4-5 (pp. 205 - 242)

Section 9-5 (pp. 580 - 585)

Section 12 - 6 (pp. 801 - 805)

Teacher Works All-In-One Planner and Resource Center CD-ROM
Exam View Assessment Suite
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.
Materials
TI-Nspire graphing calculators, graph paper, straight edge, gridded chart paper
Instructional Considerations
Key Vocabulary
depressed polynomial
synthetic division
long division
synthetic substitution
Binomial Theorem
Remainder Theorem
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Finding Real and Complex Zeros of Polynomials (7–9 days)
Planning for Effective Instructional Design and Delivery
Polynomial relationships was considered a Critical Area for Algebra 2. Students should use their prior
knowledge of operations with quadratics to deepen their conceptual understanding of operations with
polynomials. A quick review on finding the zeros of a quadratic equation, characteristics of quadratic
graphs, etc., should be the introduction to polynomials.
In this unit, students use various methods to identify and approximate the rational, real, and complex
zeros of polynomials functions. Additionally, students use zeros to construct rough graphs. Starting with
the answers to the problems, and subsequently asking students to sketch a graph based on these solutions
will support A-APR.3. For example, students should be able to take the zeros {1, 2 and 1 + i } and sketch
a graph. They must also be able to sketch a graph of the polynomial given the factors. You may want to
extend this activity and have students be able to find the factors: (x – 1)(x – 2)(x – (1 + i ))(x – (1 – i)),
multiply them together = (x – 1)(x – 2)(x2 – 2x + 2), and finally find the polynomial: y = x4 – 5x3 + 10x2 –
10x + 4. This is a fairly complicated polynomial; however, students should have the conceptual
understanding of the behavior of conjugate pairs and perfect square trinomials on the graphs (and,
therefore, the roots/solutions) of polynomials. They should also have a clear understanding of the possible
number of roots a polynomial might produce. They should be able to apply these concepts using graphs,
factors, and roots. This work will support the subsequent unit on graphing which extends the concept to
include graphing polynomial and rational functions.
This unit also is reinforces students’ previous work with the complex number system. Students will
explore operations with complex numbers, including adding, subtracting, multiplying, and dividing
complex numbers. The history of imaginary numbers is interesting, particularly for its patterns and
applications. When students have encountered the quadratic formula, the existence of such numbers is
clear. Students should first experience the idea of non-real solutions to ground their understanding of
imaginary and complex numbers. Graphical representations of quadratic functions with complex roots
visually explain that there are clearly no real zeros, but defining these roots as imaginary still takes a bit
of practice and validation. Students must have ample opportunity to explore imaginary numbers by
performing arithmetic operations with the powers and other complex forms of i. The variable-like
property of i when adding and subtracting complex numbers supports its “unknown” quality, but, unlike
most variables, i is clearly defined. This concept is fairly abstract and difficult for students to grasp.
Students will need a reminder of the powers of i. Have students make a graphic organizer to review their
knowledge. Students can use an identifying similarities and differences strategy to represent the cyclical
nature of imaginary numbers. They can make a comparison matrix to show how multiplying 1 by
itself x amount of times gives a pattern:
i0
1
i1
i
i2
–1
i3
–i
i
4
1
i
5
i
Have students show what kind of parabolic graphs are associated with complex numbers. Also, have
students make a quadratic function with complex roots, solve using conjugates, and share out their
findings with the whole class.
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Finding Real and Complex Zeros of Polynomials (7–9 days)
Precalculus, Quarter 1, Unit 1.2
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Notes
Providence Public Schools
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Finding Real and Complex Zeros of Polynomials (7–9 days)
Providence Public Schools