Polynomials

Lesson 8

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).

Foundational Standards

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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. 

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

 

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

 

Guiding Questions

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Problem 2

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

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

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

Target Task

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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.

References