Secondary Mathematics III
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Strand: ALGEBRA - Arithmetic With Polynomials and Rational Expressions (A.APR)

Perform arithmetic operations on polynomials, extending beyond the quadratic polynomials (Standard A.APR.1). Understand the relationship between zeros and factors of polynomials (Standards A.APR.2-3). Use polynomial identities to solve problems (Standards A.APR.4-5). Rewrite rational expressions (Standards A.APR.6-7).

Standard 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).

• ALGEBRA - Arithmetic With Polynomials and Rational Expressions (A.APR) - Sec Math III Core Guide
  The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics III - Arithmetic With Polynomials and Rational Expressions (A.APR).
• Graphing from Factors I
  The purpose of this task is to help students understand the relationship between the factors of a polynomial and the x-intercepts of the graph of the polynomial. By giving students two different polynomials with the same factors the task draws attention to the fact that both polynomials cross the x-axis at the same points. Students are then invited to reflect on why this is so by looking at the structure of the polynomials.
• Graphing from Factors II
  The purpose of this task is to give students an opportunity to see and use the structure of the factored form of a polynomial (MP7).
• Graphing from Factors III
  The task has students use the remainder theorem to deduce a linear factor of a cubic polynomial, and then to completely factor the polynomial. Students will need some procedure (e.g., synthetic or long division, or guess-and-check the coefficients) for determining the quadratic factor. Having the factored form permits students to deduce much about the structure of the graph.
• Introduction to the Materials (Math 3)
  Introduction to the Materials in the Mathematics Three of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• Module 3: Polynomial Functions - Student Edition (Math 3)
  The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
• Module 3: Polynomial Functions - Teacher Edition (Math 3)
  The Mathematics Vision Project, Secondary Math Three Module 3, Polynomial Functions, begins with a task that links linear, quadratic, and cubic functions together by highlighting the rates of change of each function type and using a story context to show that a linear function is the sum of a constant, a quadratic function is the accumulation or sum of a linear function, and a cubic function is the sum of a quadratic function.
• Representing Polynomials
  This lesson unit is intended to help educators assess how well students are able to translate between graphs and algebraic representations of polynomials.
• Solving a Simple Cubic Equation
  The purpose of this task is twofold. First, it prompts students to notice and explain a connection between the factored form of a polynomial and the location of its zeroes when graphed. Second, it highlights a complication that results from a seemingly innocent move that students might be tempted to make: "dividing both sides by x."
• The Missing Coefficient
  The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials.
• Zeroes and factorization of a general polynomial
  This task builds on ''Zeroes and factorization of a quadratic function'' parts I and II. The teacher may wish to recall the result from the first of these tasks, generalized to the polynomials of degree d considered here.
• Zeroes and factorization of a non polynomial function
  The level of the task is appropriate for assessment but since its intention is to provide extra depth to the standard A-APR.2 it is principally designed for instructional purposes only.
• Zeroes and factorization of a quadratic polynomial I
  For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by xr. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact.
• Zeroes and factorization of a quadratic polynomial II
  This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions.

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