# Polynomials

## Objective

Write polynomial functions from solutions of that polynomial function.

## Common Core Standards

### Core Standards

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• A.APR.B.3 — 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.

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• F.BF.A.1

## Criteria for Success

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1. Notice and describe the pattern of number of points minus 1 defining the largest degree a set of points can define.
2. Identify systems of equations as a viable process for determining the coefficients for the polynomial functions.
3. Describe that the number of points presented does not automatically determine the degree of the polynomial function.

## Tips for Teachers

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In terms of pacing, this lesson can be taught over two days.

## Anchor Problems

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### Problem 1

Sarah is given two points that lie on a line.

${f(8)=0 }$

${f(-2)=5}$

She first writes the slope intercept form of the equation:

${y=mx+b}$

Then, she substitutes one point into the equation:

${0=(8)m+b}$

She then substitutes the other point into the slope intercept form:

${5=(-2)m+b }$

What are Sarah’s next steps in finding the equation of this line?

#### References

EngageNY Mathematics Algebra II > Module 1 > Topic B > Lesson 20Example 1

Algebra II > Module 1 > Topic B > Lesson 20 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

### Problem 2

The general form for a polynomial function is:

${{a_n}x^n+a_{n-1}x^{n-1}+a_1x+a_0 ...}$

where $n$ is the degree of the polynomial, and $a_{n}$ is the leading coefficient.

So, for example: a 4th-degree polynomial has a general form of:

${ax^4+bx^3+cx^2+dx+e}$

What is the largest degree you could uniquely define if you were given three points?

### Problem 3

What is the polynomial $P$ such that $P(-1)=10$, $P(2)=1$, and $P(0)=3$

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

Use the remainder theorem to find a quadratic polynomial $P$ so that $P(1)=5$, $P(2)=12$, and $P(3)=25$. Give your answer in standard form.