What is a polynomial?
• 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).
In this section students become familiar with the arithmetic of polynomials. They add, subtract, and multiply them, and they use the properties of operations, particularly the distributive law, to express them as a sum of powers with coefficients. They recognize that every polynomial can be put in this form. They use polynomials to express and verify numerical patterns.
The emphasis in this section should not be on formal definitions or formal proofs of closure properties. Rather the emphasis is on preparing students for the manipulations they will be using the coming sections, when they start to study polynomial functions.
WHAT: Students are presented with a scenario where a student notices that the digits in the first few powers of 11 are the same as the coefficients in the first few powers of $1 + x$. They are asked to decide if this always holds true.
WHY: This task illustrates the analogy between the system of integers and the system of polynomials. In both systems you can add, subtract, and multiply; the only difference is that polynomials include one or more variables. The task helps students understand that in dealing with polynomials they can treat the variables just like any other number and can use the properties of operations they learned from arithmetic to work with polynomials.
WHAT: Students investigate Trina’s technique for generating Pythagorean triples. It is first presented as a rule that involves picking any two numbers and performing a sequence of operations with them. Students represent the rule algebraically and then verify it using a polynomial identity in two variables.
WHY: This task further illustrates the analogy between the system of integers and the system of polynomials. By starting with a description that involves picking two unspecified numbers and ending with a polynomial identity it reinforces the idea that polynomials can be manipulated as if the variables are just numbers. It also helps students see polynomial identities as generalizing a collection of numerical identities with the same structure (MP.7, MP.8).
WHAT: Students are given various polynomials and asked to decide if their values are non-negative for all values of the variable. They are then asked various questions about the structure of such polynomials, such as whether their coefficients must always be positive.
WHY: This task connects the arithmetic of polynomials with the properties of functions defined by polynomials. Coming at the end of the section, this task begins to move students towards thinking of polynomials as functions, in preparation for the next section where they will start graphing polynomials.