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Pre-Calculus Lesson Plans - Polynomials

MA Standard 2.0: Students are adept at the arithmetic of complex numbers

MA Standard 4.0: Students know the statement of and can apply the fundamental theorem of algebra

MA Standard 6.0: Students find the roots and poles of a rational function and can graph the function and locate its asymptotes

Goals: Convert a quadratic function between vertex form and standard form
Determine the end behavior and the behavior at a zero of a polynomial
Synthesize division of polynomials
Synthesize synthetic division
Derive and apply the Remainder Theorem
Derive the Factor Theorem
Synthesize the zeroes of a polynomial
Apply the rational zeros theorem and others
Complex Numbers Arithmetic
Simplify rational complex numbers and completely factor a real polynomial having real and complex roots
Stating Fundamental Theorem of Algebra
Review for test
Graph a rational function
Determine the roots and asymptotes of a rational function
Analyze rational functions with slant asymptotes
Review for test

Vocabulary: X-intercept, y-intercept, relative minimum, relative maximum, polynomial, degree of a polynomial, quadratic function, standard form, vertex form, common form of a function, completing the square, remainder theorem, factor theorem, synthetic division, rational zeros theorem, Descarte’s Rules of signs, difference of two squares, difference of two cubes, sum of two squares , sum of two cubes, complex number, real part, imaginary part, complex conjugate, rational function, vertical asymptote, horizontal asymptote.

Procedure: This long unit is divided into two parts: (1) polynomials and (2) rational expressions. We give a test at the end of each part.
We start with an introduction of polynomials, but quickly revert to quadratic expressions, prior knowledge from algebra and algebra II, as a means of teaching some of the concepts of polynomials. We also re-teach completing the square as a means of solving quadratic equations. (I prefer completing the square to teaching the quadratic formula because it is the proof of the quadratic formula and it is a method that solves all quadratics, including those that are not factorable and those that result in complex solutions. I also prefer it over factoring, which is limited in application.) We finish the lesson with applications to realistic problems.

With this lesson we introduce a new concept; the 1-minute quiz. This is a short quiz that covers some fundamental skill that is necessary for success in the unit. In this case it is solving quadratics. The quiz includes approximately 10 problems. The students are given a short time, “1-minute”, to solve. Nobody completes the quiz. The quiz is collected and correct solutions are indicated with a check. Incorrect or unattempted problems are not marked. The quiz is returned to the students during several succeeding lessons. Each time the students can correct any problem and can complete ones they haven’t previously attempted. The objective is to have the students reflect on what they got right and what they got wrong. The goal is to have all the problems correctly done by the end of the unit. The 1-minute quiz is consistent with formative assessment, a “best practice” strategy for teaching.

In lesson 2 we start to develop the properties of polynomial functions. The first, end behavior, is discovered through an investigation. We fol low with the three equivalents of a zero of a polynomial; factor, x-intercept and zero of the function. The Intermediate Value Theorem and the impact of repeated zeros are explained. We finish this lesson with a return to the 1-minute quiz.

The focus of lesson 3 is factoring and long division. This lesson is about skill development. Consequently, the focus is on worksheets. The first worksheet involves factoring and the second long division.

In lesson 4 we meticulously expand the students ’ knowledge of polynomials. We teach synthetic division, the remainder theorem and the factor theorem. Again, the focus is on practice.

The rational zeros theorem and Descarte’s Rules of Signs round off the techniques for determining the zeros of a polynomial. We use an investigation to help the students understand that the rational zeros are related to the factors of the lowest order and the highest order terms of a polynomial. The focus of Descarte’s Rules of Signs is their usage rather than an explanation.

This next lesson starts with the 1-minute quiz for this unit. We can always use the 1-minute quiz in any lesson that has time left at the end. The past few lessons are hard enough to complete without the 1-minute quiz. However, we should never abandon the quiz since it is so effective at teaching a skill. After the warm up we introduce x3 = 8. The students are asked to determine all the solutions to this equation. We encourage them by hinting that there are three solutions. In this manner, we introduce complex numbers. We then review the factorization of the difference of two squares, the sum of two squares, the difference of two cubes and the sum of two cubes. Over time, we’ve expanded the time on this topic as we found that weaknesses in algebraic skills on topics such as these hamper our students in calculus.

The warm up is the 1-minute quiz. Lesson 7 is a standard lesson on complex algebra. Students are encouraged to recall this information from algebra and algebra II. However, we still cover the basics and provide ample practice on all algebraic operations. With the explanation of complex numbers, the students are ready for the Fundamental Theorem of Algebra.

At this point we provide a review day. This is a very large unit and to go to its completion before assessing the students puts a huge load on their memory. Furthermore, rational functions are sufficiently different to justify a gap in the presentations.

Rational functions are organized about the degree of the polynomial in the denominator. First we study linear polynomials and then quadratic. Once we finish these two topics we study slant asymptotes, which relates to the degree of the numerator. We introduce the concept of asymptote through the elementary function 1/x. This investigation allows the student to formulate his own interpretation of asymptote. We then use functional transformations to explain asymptotes that are not x = 0 and y = 0. This short topic is spread over three lessons to provide time for the students to absorb the new ideas.

Relevant Resource Material
• KBAC Extreme Value Word Problems
• KBAC Division
• KBAC Polynomial Zeros
• KBAC Complex Numbers
• KBAC Complex Number Arithmetic
• KBAC Rational Functions

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