# Polynomials

## 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).

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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${f(x)=x^3-9x}$.

#### 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?

### 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?

## 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.

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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.

${18x^4-32}$