Mastering Polynomial Long Division

Algebra 2 Grades High School 3:29 Video
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Lesson Description

Learn to divide polynomials using long division with step-by-step examples. This lesson covers setting up the problem, dividing, subtracting, and handling remainders.

Video Resource

Dividing Polynomials Using Long Division

Mario's Math Tutoring

Duration: 3:29
Watch on YouTube

Key Concepts

  • Polynomial Long Division
  • Dividend, Divisor, Quotient, Remainder
  • Subtracting Polynomials
  • Distributing Terms
  • Remainders as Fractions

Learning Objectives

  • Students will be able to set up and perform polynomial long division.
  • Students will be able to identify the quotient and remainder in polynomial long division.
  • Students will be able to express the remainder as a fraction of the divisor.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing basic long division with numbers to draw a parallel to polynomial long division. Explain the parts of a division problem (dividend, divisor, quotient, remainder).
  • Example 1: (4x^2 - 2x + 3) / (x - 1) (10 mins)
    Follow the video's first example, explaining each step clearly: 1. Set up the long division problem. 2. Determine what to multiply the divisor (x - 1) by to get the first term of the dividend (4x^2). 3. Multiply and subtract (remember to change signs). 4. Bring down the next term. 5. Repeat until no more terms can be brought down. 6. Express the remainder as a fraction over the divisor.
  • Example 2: (6x^2 - 3x + 9) / (3x - 1) (15 mins)
    Work through the second example, emphasizing the importance of lining up like terms and being careful with subtraction: 1. Set up the long division problem. 2. Determine what to multiply the divisor (3x - 1) by to get the first term of the dividend (6x^2). 3. Multiply and subtract (remember to change signs). 4. Bring down the next term. 5. Repeat until no more terms can be brought down. 6. Express the remainder as a fraction over the divisor. Note the fractional coefficient.
  • Practice Problems (15 mins)
    Provide students with practice problems to solve individually or in pairs. Circulate to provide assistance as needed. Example problems: (x^2 + 5x + 6) / (x + 2) (2x^2 - 7x + 3) / (x - 3) (x^3 - 1) / (x - 1)
  • Wrap-up (5 mins)
    Review the key steps of polynomial long division and answer any remaining questions. Preview the next lesson on synthetic division.

Interactive Exercises

  • Online Polynomial Long Division Calculator
    Students can use an online calculator to check their work after completing problems manually. This allows for immediate feedback and reinforces correct procedures.
  • Error Analysis
    Present students with worked-out examples of polynomial long division that contain common errors. Have them identify and correct the mistakes.

Discussion Questions

  • How is polynomial long division similar to numerical long division?
  • What happens if a term is missing in the dividend (e.g., x^3 + 1)?
  • Why is it important to change the signs when subtracting in long division?
  • How can you check your answer after performing polynomial long division?

Skills Developed

  • Algebraic Manipulation
  • Problem Solving
  • Attention to Detail
  • Critical Thinking

Multiple Choice Questions

Question 1:

What is the first step in polynomial long division?

Correct Answer: Set up the long division problem

Question 2:

When subtracting polynomials in long division, what must you remember to do?

Correct Answer: Change the signs of the second polynomial

Question 3:

What do you do with the remainder in polynomial long division?

Correct Answer: Write it as a fraction over the divisor

Question 4:

When dividing (x^2 + 3x + 2) by (x + 1), what is the first term of the quotient?

Correct Answer: x

Question 5:

What is the remainder when (x^2 + 5x + 6) is divided by (x + 3)?

Correct Answer: 0

Question 6:

What is the quotient when (2x^2 + x - 3) is divided by (x - 1)?

Correct Answer: 2x + 3

Question 7:

In polynomial long division, the degree of the remainder must be __________ than the degree of the divisor.

Correct Answer: less

Question 8:

Which of the following is equivalent to (x^3 - 8) / (x - 2)?

Correct Answer: x^2 + 2x + 4

Question 9:

If the divisor is (x + 2) and the quotient is (x - 3), what polynomial was divided (ignoring any remainder)?

Correct Answer: x^2 - x - 6

Question 10:

When dividing (4x^3 - 2x + 5) by (2x - 1), what is the leading term of the quotient?

Correct Answer: 2x^2

Fill in the Blank Questions

Question 1:

The polynomial being divided is called the _________.

Correct Answer: dividend

Question 2:

The polynomial that you are dividing by is called the _________.

Correct Answer: divisor

Question 3:

The result of the division (before the remainder) is called the _________.

Correct Answer: quotient

Question 4:

If the remainder is zero, the divisor is a _________ of the dividend.

Correct Answer: factor

Question 5:

Subtraction is the same as adding the _________.

Correct Answer: opposite

Question 6:

When setting up long division, make sure to line up _________ terms.

Correct Answer: like

Question 7:

The remainder is written as a fraction with the remainder as the numerator and the _________ as the denominator.

Correct Answer: divisor

Question 8:

If a term is missing in the dividend (e.g., x^3 + 1), use a _________ as a placeholder.

Correct Answer: zero

Question 9:

Polynomial long division is used to simplify rational expressions and solve polynomial _________.

Correct Answer: equations

Question 10:

The degree of the remainder must always be less than the degree of the _________.

Correct Answer: divisor

Educational Standards

A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions. A2.A.1.4:Solve polynomial equations with real roots using various methods (e.g., polynomial division, synthetic division, using graphing calculators or other appropriate technology).

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