Mastering Polynomial Long Division: A Step-by-Step Guide

Algebra 2 Grades High School 8:18 Video
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Lesson Description

Learn polynomial long division with clear, step-by-step examples and strategies to simplify the process. This lesson covers setting up the problem, handling missing terms, and interpreting remainders.

Video Resource

How to do Polynomial Long Division Step by Step

Mario's Math Tutoring

Duration: 8:18
Watch on YouTube

Key Concepts

  • Polynomial long division
  • Dividend, divisor, quotient, and remainder
  • Placeholders for missing terms

Learning Objectives

  • Set up a polynomial long division problem correctly.
  • Perform the steps of polynomial long division accurately.
  • Identify and handle missing terms using placeholders.
  • Express the result of the division including the quotient and remainder.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing basic long division with numbers to draw a parallel to polynomial long division. Introduce the concept of dividing polynomials and explain its importance in simplifying expressions and solving equations. Briefly mention the video by Mario's Math Tutoring as a helpful resource.
  • Example 1: Basic Polynomial Long Division (10 mins)
    Work through the first example from the video (x^2 - 3x + 5 divided by x - 2) step-by-step. Emphasize lining up terms, subtracting (or adding the opposite), and bringing down the next term. Explain how to determine what to multiply the divisor by to match the leading term of the dividend.
  • Example 2: Handling Missing Terms (15 mins)
    Tackle the second example (2x^3 - 2x^2 + 7 divided by x^2 + 1). Highlight the importance of using placeholders (0x) for missing terms. Stress how these placeholders help to keep the problem organized and prevent errors. Walk through the distribution and subtraction steps carefully.
  • Example 3: Zero Remainder (10 mins)
    Complete the third example from the video. This example illustrates a case where the remainder is zero. Discuss what it means when the remainder is zero, connecting it to the concept of factors.
  • Practice and Review (10 mins)
    Assign practice problems similar to the examples covered in the video. Encourage students to work independently and then review the solutions together as a class.

Interactive Exercises

  • Error Analysis
    Present students with partially completed polynomial long division problems with errors. Ask them to identify the mistakes and correct them.
  • Partner Practice
    Have students work in pairs to solve polynomial long division problems. One student can perform the division while the other checks their work and offers assistance.

Discussion Questions

  • Why is it important to use placeholders for missing terms in polynomial long division?
  • How is polynomial long division similar to and different from regular long division with numbers?
  • What does it mean if the remainder is zero after performing polynomial long division?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Attention to detail

Multiple Choice Questions

Question 1:

What is the first step in performing polynomial long division?

Correct Answer: Divide the leading term of the dividend by the leading term of the divisor

Question 2:

What should you do if a term is missing in the dividend (e.g., no x term)?

Correct Answer: Use a placeholder with a coefficient of zero (e.g., 0x)

Question 3:

When do you stop the long division process?

Correct Answer: When the degree of the remainder is less than the degree of the divisor

Question 4:

In polynomial long division, what does the term 'divisor' represent?

Correct Answer: The polynomial we are dividing by

Question 5:

Which expression represents the correct setup for (3x^3 + 2x - 5) ÷ (x + 1) using placeholders?

Correct Answer: (3x^3 + 0x^2 + 2x - 5) ÷ (x + 1)

Question 6:

What does it mean if the remainder is zero after polynomial long division?

Correct Answer: The divisor is a factor of the dividend.

Question 7:

If you divide (x^2 + 5x + 6) by (x + 2), what is the quotient?

Correct Answer: x + 3

Question 8:

Which of the following represents the correct structure for expressing the answer to a polynomial long division problem?

Correct Answer: Quotient + (Remainder / Divisor)

Question 9:

After dividing (x^3 - 8) by (x - 2), what is the value of the remainder?

Correct Answer: 0

Question 10:

In the expression (x^2 - 4x + 3) / (x - 1), what is the dividend?

Correct Answer: x^2 - 4x + 3

Fill in the Blank Questions

Question 1:

In polynomial long division, 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:

If a term is missing in the dividend, you should use a ________ with a coefficient of zero.

Correct Answer: placeholder

Question 4:

After performing the division, the result is expressed as the ________ plus the remainder over the divisor.

Correct Answer: quotient

Question 5:

When the degree of the ________ is less than the degree of the divisor, you stop the division process.

Correct Answer: remainder

Question 6:

If the remainder is ________, it means the divisor is a factor of the dividend.

Correct Answer: zero

Question 7:

Subtraction in polynomial long division can be simplified by changing the signs and ________.

Correct Answer: adding

Question 8:

In the division problem (x^3 + 2x^2 - x + 5) / (x - 1), the leading term of the dividend is ________.

Correct Answer: x^3

Question 9:

The terms in both the divisor and dividend should be in ________ order based on the exponent of the variable.

Correct Answer: descending

Question 10:

The result of the division is known as the ________, indicating how many times the divisor is contained within the dividend.

Correct Answer: quotient

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).

Teaching Materials

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