Page 42 - IM_Algebra2_FL Print Sample.pdf

work as they investigate polynomials of higher degree and the features that all polynomial functions have in common.
The unit begins with an introduction to some situations polynomial functions are good at modeling. Students learn to identify the degree of a polynomial and gain exposure to what graphs of polynomials can look like as they build intuition for what features these graphs can and cannot have.
Focusing on functions represented by expressions written in factored form and also represented by graphs, students make connections between linear factors and horizontal intercepts, identifying that a factor of means is a zero of the function and is a horizontal intercept. The eVect of the degree and leading coeUcient on end behavior is established along with the eVect of multiplicity on the shape of the graph near zeros of the function. Taking in all of these features, students learn to make rough sketches of polynomial functions from expressions written as products of linear factors.
Embedded throughout the Srst half of the unit are opportunities for students to practice multiplying polynomials. This practice is meant to help pave the way for understanding division, which in this unit focuses on dividing a polynomial written in standard form by a known factor for the purposes of rewriting the equation in factored form. From there, the connection between division and multiplication equations is used to establish the Remainder Theorem. This allows the declaration that not only does a known factor of the form mean a polynomial has a zero at , but that if a polynomial has a zero at
, then it must also have as a factor.
Students transition to working with rational functions by considering situations they model, such as average cost. The asymptotic behavior of their graphs is examined as it relates to the structure of the equation. Also building on structure, students employ polynomial division to rewrite rational expressions for the purpose of identifying the end behavior of the function. Students then spend several lessons focusing on solving rational equations and making sense of how the process can lead to possibile solutions that are in fact not solutions (so-called extraneous solutions).
In the Snal section students study polynomial identities. They hone skills manipulating polynomial expressions while proving, or disproving, that two expressions are equivalent. Lastly, students return to geometric sequences Srst examined in the previous unit and, using polynomial identities, derive the formula for the sum of the Srst terms in a geometric sequence and then use the formula to solve problems.
39