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Subject: High School Mathematics Unit Overview Course: Algebra II Unit: #1 Polynomials & Rational Expressions Instructional Time: 3 Weeks Essential Question(s) 1. What is a “zero” of a polynomial? 2. How can zeros of polynomials be interpreted? 3. How can coefficients for the Binomial Theorem be found? 4. What is the Fundamental Theorem of Algebra? 5. What is the connection between Rational Expressions and Polynomials? Big Idea(s) If p(a) = 0 then (x – a ) is a factor of polynomial p(x). The zeros of a polynomial are the x-intercepts of its graph. Pascal’s Triangle can be used to easily find Binomial Coefficients. The number of possible complex zeros of a polynomial can be determined by its degree. Rational Expressions are formed with Polynomial numerators and denominators. © 2013 The K-12 Curriculum Project, Inc. Indicates a modeling standard linking mathematics to everyday life, work, and decision-making (+) Indicates additional mathematics to prepare students for advanced courses. Page 1 Unit Overview Subject: High School Mathematics Course: Algebra II Unit: #1 Polynomials & Rational Expressions Instructional Time: 3 Weeks Priority Standards A-APR2. 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-APR3. 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. A-APR5. (+) 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. A-APR6. Rewrite 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 of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. N-CN9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. © 2013 The K-12 Curriculum Project, Inc. Indicates a modeling standard linking mathematics to everyday life, work, and decision-making (+) Indicates additional mathematics to prepare students for advanced courses. Page 2 Unit Overview Subject: High School Mathematics Course: Algebra II Unit: #1 Polynomials & Rational Expressions Instructional Time: 3 Weeks Priority Standard Connection Supporting Standards A-APR1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-APR3 A-APR5 A-APR4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. A-APR3 A-APR7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-APR6 N-CN8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). © 2013 The K-12 Curriculum Project, Inc. Indicates a modeling standard linking mathematics to everyday life, work, and decision-making (+) Indicates additional mathematics to prepare students for advanced courses. Page 3 A-APR3 N-CN9 Unit Overview Subject: High School Mathematics Course: Algebra II Unit: #1 Polynomials & Rational Expressions Instructional Time: 3 Weeks Analysis of Priority Standards A-APR2. 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). Action Bloom Idea DOK Know Knowledge Polynomial, Remainder, Division of Polynomials 1 Apply Application P(a) = 0 if and only if (x-a) is a factor 3 A-APR3. 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. Action Bloom Idea DOK Identify Knowledge Zeros of Polynomials 1 Use Application Construct a graph 2 n A-APR5. (+) Know and apply the Binomial Theorem for the expansion of (x + y) 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. Action Bloom Idea DOK n Know Knowledge Binomial Theorem and expansion of (x + y) 1 Apply Application Coefficients determined by Pascal’s Triangle 3 A-APR6. Rewrite 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 of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Action Bloom Idea DOK Rewrite Synthesize Different forms of rational expressions 2 Use Application Inspection, long division 2 N-CN9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Action Bloom Idea DOK Know Knowledge The Fundamental Theorem of Algebra 2 Show Evaluation Quadratic Polynomials 2 © 2013 The K-12 Curriculum Project, Inc. Indicates a modeling standard linking mathematics to everyday life, work, and decision-making (+) Indicates additional mathematics to prepare students for advanced courses. Page 4 Unit Overview Subject: High School Mathematics Course: Algebra II Unit: #1 Polynomials & Rational Expressions Instructional Time: 3 Weeks © 2013 The K-12 Curriculum Project, Inc. Indicates a modeling standard linking mathematics to everyday life, work, and decision-making (+) Indicates additional mathematics to prepare students for advanced courses. Page 5