Polynomials

Lesson 6

Objective

Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.

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.

  • F.BF.A.1.B — Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Foundational Standards

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  • A.REI.A.1

  • A.REI.B.3

Criteria for Success

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  1. Add like terms in polynomials. 
  2. Distribute the subtraction sign across polynomials and subtract terms accurately. 
  3. Predict the end behavior of sum or difference based on the leading coefficients of the polynomials. 
  4. Determine when a polynomial difference will result in a lesser-degree polynomial. 
  5. Use [VARS] "Y-VARS" "1. Function" to input Y1, Y2, Y3, etc., to graph the sum and difference of two functions 
  6. Graph multiple functions in [Y=] and [GRAPH]. "Turn on" and "turn off" functions in [Y=].

Anchor Problems

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Problem 1

If $${f(x)= x^3-3x^2+2 }$$

 

and 


$${g(x)=-4(x+2)^2-5}$$

 

What is $${ f(x)+g(x)}$$
What is $${ f(x)-g(x)}$$
What is $${g(x)-f(x)}$$

Guiding Questions

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Problem 2

Is the following statement always, sometimes, or never true? 
“The difference between two polynomials will be the same degree as the highest-degree polynomial in the difference.”

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 write the original polynomials that result in a particular end behavior. 
  • Include problems where there are missing coefficients or exponents that when added or subtracted result in a particular end behavior. 
  • Include problems that confirm that polynomial addition is commutative but polynomial subtraction is not—following from properties of rational numbers.

Target Task

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Find $${h(x)-j(x)}$$.

$${h(x)=5x^4-3x^2+4}$$

$${j(x)=2x^3+5x^4+4}$$

  • What is the end behavior of the initial polynomials? 
  • What is the end behavior of the resultant polynomial? 
  • In what cases will a sum or difference of two polynomials with the same end behavior result in a polynomial with different end behavior?