# Modeling with polynomials

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

The main focus of modeling in this course is situations that can be modeled by linear, exponential, and quadratic functions. However, some contexts naturally give rise to polynomial models. In this section students make use of all that they have learned about polynomial functions to solve problems in such contexts.

## Tasks

WHAT: Students are given a situation in which a bank account has various different deposits every year for 4 years. They construct a polynomial function that models the balance in a bank account at the end of the 4 years as a function of the annual growth factor. Then they find what the growth factor must be to give a particular balance by graphically approximating the solution to a polynomial equation. Finally, they generalize the situation to a larger number of years with a higher degree polynomial.

WHY: The purpose of this task is to show a context where a polynomial function arises from a real world context and there is a need to solve a polynomial equation.

## External Resources

#### Description

WHAT: Students are given a situation in which they are bidding at an auction against another buyer. The auction might result in a profit, taking into account the value of the item, if they win it for a bid that is less than its value. They construct a quadratic function that models their expected profit and graph it to find the bid that maximizes their expected profit (F-IF.B.4$^\star$). They then repeat the calculation with more competing buyers, resulting in higher degree polynomials.

WHY: This lesson provides a context in which a polynomial model is useful in finding an optimal solution. By interpreting a graph, students can find an optimal solution. Significant quantitative reasoning (MP.2) is required to derive the model (MP.4). Note: students must apply previously learned skills in probability, such as computing the probability of compound events and modeling the expected value of an auction bid. This lesson may not be appropriate for students without the prerequisite probability knowledge.

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#### Standards

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