# Polynomials

## Objective

Divide polynomials by binomials to determine linear factors.

## Common Core Standards

### Core Standards

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• A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

• A.APR.B.2 — Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

• 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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• A.REI.A.1

• A.REI.B.3

## Criteria for Success

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1. Describe that just like with the operations of multiplication, addition, and subtraction, polynomial division follows the same principles as integer division.
2. Divide binomials into polynomials using long division.
3. Identify that when the remainder of this division is zero, the binomial is a factor of the polynomial.

## Tips for Teachers

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In this lesson, students will focus on long division and not be introduced to synthetic division. For more information about reasoning for this, see the Algebra Progressions for the Common Core, page 9. Lane Walker also provides some additional reasoning on the topic. She addresses the topic again in this post.

## Anchor Problems

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### Problem 1

Find the product.

${(x-3)(x^2-2x+3)}$

Once we find the product, how would we work backwards to get both factors again?

#### Guiding Questions

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### Problem 2

What is the other factor you would multiply ${x-2}$ by to get the polynomial ${x^4-4x^2-5x+10}$?

#### Guiding Questions

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## 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.

• Include problems where students are given the graph of the polynomial function (with an integer root), the equation, and they need to identify one linear factor from the root and then show this as a factor through long division.
• Include problems where students are given the roots and they need to test one of the factors through long division.

## Target Task

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### Problem 1

One factor of the polynomial ${2x^4+2x^3+3x+3}$ is ${(x+1)}$. What is the other factor?

### Problem 2

How can you confirm that ${x+1}$ is indeed a factor of ${2x^4+2x^3+3x+3}$ through the long division?