Polynomials

Lesson 9

Objective

Identify and factor with difference of two squares in quadratic and quartic polynomials. Describe identity of difference of two squares. Describe the zeros that represent the resultant factors.

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.

  • A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

Foundational Standards

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  • F.IF.C.8.A

Criteria for Success

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  1. Identify the difference of two squares in fourth-degree as well as second-degree polynomials. 
  2. Describe an identity as a rule that always works and may also be known as a factoring pattern. 
  3. Identify all of the major factoring patterns of difference of two squares.
  4. Determine that the sum of two squares will result in a function with complex solutions. 
  5. Identify greatest common factors from polynomials.

Anchor Problems

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Problem 1

Find the zeros of the function$${f(x)=x^3-9x}$$.

Guiding Questions

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References

Illustrative Mathematics Solving a Simple Cubic Equation

Solving a Simple Cubic Equation, accessed on Sept. 25, 2017, 2 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

What are all of the linear factors of $${2x^4-162}$$? How would you write this expression in factored form?

Guiding Questions

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Problem 3

How are the roots of a sum of two squares, such as $${x^2+9}$$, different from the roots of a difference of two squares?

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 error analysis problems with factoring polynomials.

Target Task

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Problem 1

How do the following problems represent differences of two squares? Factor to show the relationships. 

$${4x^4-1}$$


$${x^{10}-36}$$

Problem 2

Factor the following functions and name all of the real and complex zeros. 

$${18x^4-32}$$