# Ch 6: Common Core HS Algebra: Polynomials

### About This Chapter

**Standards** 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. (CCSS.Math.Content.HSA-APR.A.1)

**Standards** 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*). (CCSS.Math.Content.HSA-APR.B.2)

**Standards** 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. (CCSS.Math.Content.HSA-APR.B.3)

**Standards** Prove polynomial identities and use them to describe numerical relationships. (CCSS.Math.Content.HSA-APR.C.4)

**Standards** Know and apply the Binomial Theorem for the expansion of (*x* + *y*)^*n* in powers of *x* and *y* for a positive integer *n*, where *x* and *y* are any numbers, with coefficients determined for example by Pascal's Triangle. (CCSS.Math.Content.HSA-APR.C.5)

**Standards** Rewrite simple rational expressions in different forms; write *a*(*x*)/*b*(*x*) in the form *q*(*x*) + *r*(*x*)/*b*(*x*), where *a*(*x*), *b*(*x*), *q*(*x*), and *r*(*x*) are polynomials with the degree of *r*(*x*) less than the degree of *b*(*x*), using inspection, long division, or, for the more complicated examples, a computer algebra system. (CCSS.Math.Content.HSA-APR.D.6)

## About This Chapter

Students who have mastered this standard will be able to add, subtract and multiply polynomials with ease. They will also be able to divide polynomials using long or synthetic division. Those who complete this chapter will develop enough skills to comfortably rewrite some rational expressions.

Lessons in this standard cover concepts such as:

- Defining what a polynomial is
- Understanding concepts related to polynomial identities and combinatorics
- Using the binomial theorem and remainder theorem
- Multiplying binomials
- Constructing graphs using the zeros of polynomials

Students demonstrate mastery of these standards when they can explain the relationship between the factors and zeros of polynomials, perform arithmetic operations on polynomials and solve problems with polynomial identities. Mastering these standards can help support preparedness for any future course or career that deals with solving problems that feature unknown variables.

### How to Use These Lessons in Your Classroom

Here are some tips for how to use the lessons to support instruction in these standards:

#### Adding, Subtracting, Multiplying and Dividing Polynomials Lessons

Develop a worksheet that covers adding, subtracting, multiplying and dividing polynomials. Display the video lessons covering the concepts from the worksheet. Assign the worksheet for the students to complete for homework. Before your students turn in the assignment, go over each problem to make sure they understand each type of operation.

#### Pre-Quiz and Post-Quiz Lessons

In order to highlight the important concepts found in this chapter, have your students take the lesson quizzes before diving into the chapter content. Spend the next few classes investigating the concepts found in the chapter lessons. After the students finish the lessons, require them to complete the quizzes to assess their information retention.

#### Constructing a Graph by Using Zeros Lessons

Create a worksheet requiring students to find the zeros of polynomials and show the video lessons 'Finding Zeros of Polynomials by Factoring' and 'Using Zeros to Construct a Graph.' Have the students complete the worksheet for homework. During the next class, pass out graph paper and require the students to graph the zeroes that they found on the worksheet.

### 1. What are Polynomials, Binomials, and Quadratics?

What do polynomials, binomials, and quadratics have in common? What are their differences? How can you identify each of them? Watch this video lesson to find the answers.

### 2. How to Add, Subtract and Multiply Polynomials

Adding, subtracting and multiplying polynomials are, basically, the same as adding, subtracting and multiplying numbers. They only difference is that we have a pesky variable to worry about, but this video will show you that's no problem, so no worries! This method has worked for many of my students, and I think it will work for you, too!

### 3. How to Divide Polynomials with Long Division

Arithmetic long division and polynomial long division are very similar. Yes, it's a long process, but once you have the rhythm you will get every problem correct!

### 4. How to Use Synthetic Division to Divide Polynomials

Synthetic division is a 'short-cut' way of dividing a polynomial by a monomial. You still need to know long division, sorry, but this method is way fun when you're dividing by a monomial!

### 5. Dividing Polynomials with Long and Synthetic Division: Practice Problems

Let's look at some more polynomial division problems. We will use long division and synthetic division, but this time we will have a couple of more involved problems. So, get out some paper and a pencil and let's begin!

### 6. Remainder Theorem & Factor Theorem: Definition & Examples

In this lesson, you will learn about the remainder theorem and the factor theorem. You will also learn how to use these theorems to find remainders and factors of polynomials.

### 7. What is the Binomial Theorem?

While the F.O.I.L. method can be used to multiply any number of binomials together, doing more than three can quickly become a huge headache. Luckily, we've got the Binomial Theorem and Pascal's Triangle for that! Learn all about it in this lesson.

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### Other Chapters

Other chapters within the Common Core Math - Algebra: High School Standards course

- Common Core HS Algebra: Algebraic Expressions
- Common Core HS Algebra: Linear Equations
- Common Core HS Algebra: Linear Inequalities
- Common Core HS Algebra: Quadratics
- Common Core HS Algebra: Exponents and Exponential Functions
- Common Core HS Algebra: Rational and Radical Expressions and Equations
- Common Core HS Algebra: Systems of Equations
- Common Core HS Algebra: Matrices
- Common Core HS Algebra: Sequences and Series
- Understanding Common Core Standards