# Polynomials and Rational Functions

## •  Add, subtract, and multiply polynomials and express them in standard form using the properties of operations (A-APR.A.1). •  Prove and make use of polynomial identities (A-APR.C.4). •  Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior (A-APR.B.3$^\star$, F-IF.C.7c$^\star$). •  Use the remainder theorem to find factors of polynomials (A-APR.B.2, A-APR.D6). •  Use various strategies including graphing and factoring to solve problems in contexts that can be modeled by polynomials in one variable. •  Build a rational function that describes a relationship between two quantities (F-BF.A.1). •  Graph rational functions, interpret features of the graph in terms of a context, and use the graphs to solve problems (A-SSE.A.1a, A-REI.D.11$^\star$, F-IF.B.4, F-IF.C.7d(+)). •  Express rational functions in different forms to see different aspects of the situation they model (A-APR.D.6). •  Solve simple rational equations and understand why extraneous roots can arise (A-REI.A.2).

Coming into this unit, students understand that one can do arithmetic on quadratic expressions, and have generalized that understanding to polynomial expressions. They have solved quadratic equations with complex solutions using the methods of factoring, completing the square, and the quadratic formula. They have graphed quadratic functions and understand the relationship between zeros and factors of quadratics. They are familiar with modeling situations from which quadratic relations arise.

Students have modeled contexts with simple rational functions. They have graphed these functions and interpreted features of the graphs in context, including vertical asymptotes and end behavior. They can relate the domain of a function to its graph.

In this unit, students extend their previous work with quadratics and polynomials (in unit A4) to achieve a more general understanding of polynomials. They work with polynomials as a system where you can add, subtract, and multiply, analogous to the integers. They graph polynomial functions and they understand and use the relationship between factors, zeros, and intercepts on the graph. They model with polynomial functions.

Students study the graphs of simple rational functions. They consider contexts which can be modeled with rational functions and interpret vertical and horizontal asymptotes in terms of the context. They rewrite simple rational expressions in different forms to see different aspects of the context, and they find approximate solutions using graphical methods to rational equations that arise from the context. They also solve simple rational equations algebraically and study how and why extraneous solutions may arise.

After this unit, students going into STEM fields will see more examples, including polynomial approximations to other functions. Power series expansions (like Taylor series) show up in calculus. The characteristic polynomial of a matrix is a useful tool in university level linear algebra. Students going into higher mathematics will encounter polynomials as objects that help form abstract algebraic structures such as rings and fields.

Students may encounter simple rational relationships in real life. If they take a calculus course with a rigorous algebra component, they should do more work with more complicated rational expressions and equations beforehand.

## Sections

A2.4.0 Pre-unit diagnostic assessment

#### Summary

Assess students’ ability to
• graph a quadratic by factoring to find zeros, and correctly interpreting end behavior (A-APR.B.3, F-IR.C.7c$^\star$);
• solve a quadratic equation with a method suitable to the given form of the equation (A-REI.B.4b);
• write a simple rational equation that models a situation A-CED.A.1$^\star$, and use the equation to solve problems (A-REI.A.2).

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A2.4.1 What is a polynomial?

#### Summary

• Add, subtract, and multiply polynomials and express them in standard form using the properties of operations (A-APR.A.1).
• Prove and make use of polynomial identities (A-APR.C.4).

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A2.4.2 Graphing polynomials

#### Summary

• Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior (A-APR.B.3, F-IF.C.7c$^\star$).
• Use the remainder theorem to find factors of polynomials (A-APR.B.2, A-APR.D.6).

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A2.4.3 Modeling with polynomials

#### Summary

Use various strategies including graphing and factoring to solve problems in contexts that can be modeled by polynomials in one variable.

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A2.4.4 Rational Functions

#### Summary

• Build a rational function that describes a relationship between two quantities (F-BF.A.1$^\star$).
• Graph rational functions (A-SSE.A.1a$^\star$, F-IF.C.7d$^\star$).
• Interpret the graph of a rational function in terms of a context (F-IF.B.4$^\star$).

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A2.4.5 Modeling with rational functions

#### Summary

• Graph rational functions and use the graphs to solve problems (A-REI.D.11$^\star$).
• Express rational functions in different forms to see different aspects of the situation they model (A-APR.D.6).
• Solve simple rational equations and understand why extraneous roots can arise (A-REI.A.2).

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A2.4.6 End Assessment

#### Summary

Assess students’ ability to
• graph a polynomial by factoring to find zeros, and correctly interpreting end behavior (A-APR.B.3, F-IF.C.7c$^\star$);
• apply the remainder theorem to solve a mathematical problem (A-APR.B.2);
• model with a polynomial (A-CED.A.1$^\star$), and use the model to solve a problem (A-REI.D.11$^\star$);
• rewrite a rational expression in a different form to solve a problem (A-REI.D.11$^\star$);
• write a rational equation that models a situation (A-CED.A.1$^\star$), and use the equation to solve problems (A-REI.A.2);
• demonstrate understanding that when solving a rational equation, extraneous roots may emerge as a result of the solution process (A-REI.A.2).

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