# Polynomials

## Objective

Determine if a binomial is a factor of a polynomial using the remainder theorem.

## Common Core Standards

### Core Standards

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• 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).

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• A.REI.D.10

## Criteria for Success

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1. Use the elementary understanding that if a division problem does not have a remainder, then the divisor is a factor of the dividend.
2. Describe the remainder theorem as the concept that if the division of two polynomials results in a remainder of zero, then the polynomial that acts as the divisor is a factor of the polynomial that acts as a dividend.
3. Know that if a binomial is a factor of a polynomial, then that binomial can be used to find a zero of that polynomial using the zero product property.
4. Find missing values in a polynomial using the zero product property and the remainder theorem.

## Anchor Problems

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### Problem 1

Which of the following binomials are factors of the polynomial? Explain your reasoning.

${x^4+5x^3-x^2-17x+12}$

${(x-1) }$          ${(x+1)}$          ${(x+3)}$          ${(x+5)}$          ${(x+4)}$          ${(x-2)}$

### Problem 2

What is the value of a in the polynomial shown below if ${x+2}$ is a factor of that polynomial?

${3x^4+6x^3+ax^2+3x+9}$

## 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 need to identify if the given values are roots and then identify the associated linear binomial that is a factor.

Is ${x+1}$ a factor of ${2x^2-3x-5}$? How do you know? Show your reasoning using long division and by using the value of $x$ at the root.