Polynomial Long Division: Mastering the Algorithm
Lesson Description
Video Resource
Key Concepts
- Polynomial long division algorithm
- Dividend, divisor, quotient, and remainder
- Expressing remainders as fractions
Learning Objectives
- Students will be able to perform polynomial long division accurately.
- Students will be able to express the result of polynomial long division, including the remainder, in the correct form.
- Students will be able to apply polynomial long division to simplify rational expressions.
Educator Instructions
- Introduction (5 mins)
Begin by reviewing numerical long division as demonstrated in the video (0:10-0:40). Connect this to the concept of dividing polynomials. Emphasize the importance of organization and place value (powers of x). - Introductory Example (15 mins)
Work through the first example from the video (0:40-2:49) step-by-step. Focus on aligning terms with the same degree, changing the signs during subtraction, and bringing down the next term. Explain the concept of the remainder and how to express it as a fraction over the divisor. - Example 1: Complex Polynomial Division (20 mins)
Solve the second example from the video (2:49-end) together. Emphasize the need for careful calculations and paying attention to signs. Highlight how to deal with fractional coefficients when they arise. Discuss the concept of the quotient as the 'answer' to the division problem. - Independent Practice (15 mins)
Provide students with 2-3 practice problems of varying complexity. Encourage them to check their work by multiplying the quotient and divisor and adding the remainder to see if they obtain the original dividend. - Wrap-up and Q&A (5 mins)
Summarize the key steps in polynomial long division. Answer any remaining questions and provide a brief overview of how this skill will be used in future lessons (e.g., finding roots of polynomials, simplifying rational expressions).
Interactive Exercises
- Error Analysis
Present students with worked-out examples of polynomial long division that contain errors. Have them identify the errors and correct them. - Polynomial Puzzle
Give students a polynomial dividend and a polynomial quotient. Ask them to find the divisor and remainder. This reinforces the relationship between the parts of the division problem.
Discussion Questions
- How is polynomial long division similar to numerical long division?
- What does the remainder represent in polynomial long division?
- How can polynomial long division be used to factor polynomials?
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: Divide the leading term of the dividend by the leading term of the divisor
Question 2:
When do you stop the polynomial long division process?
Correct Answer: When the degree of the remainder is less than the degree of the divisor
Question 3:
Which of the following is the correct way to write the answer to a polynomial long division problem?
Correct Answer: Quotient + (Remainder / Divisor)
Question 4:
What is the quotient when (x^2 - 4) is divided by (x - 2)?
Correct Answer: x + 2
Question 5:
What is the remainder when (x^3 + 1) is divided by (x + 1)?
Correct Answer: 0
Question 6:
When performing polynomial long division, it is important to:
Correct Answer: Align terms with the same degree
Question 7:
In the division problem (x^3 - 8) / (x - 2), what is the dividend?
Correct Answer: x^3 - 8
Question 8:
Why is it important to change the signs and add during polynomial long division?
Correct Answer: To avoid getting a negative remainder
Question 9:
What does it mean if the remainder of a polynomial long division problem is zero?
Correct Answer: The divisor is a factor of the dividend.
Question 10:
If you're dividing a polynomial of degree 5 by a polynomial of degree 2, what is the degree of the quotient?
Correct Answer: 3
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 polynomial long division is called the ___________.
Correct Answer: quotient
Question 4:
The leftover amount after polynomial long division is called the ___________.
Correct Answer: remainder
Question 5:
When setting up polynomial long division, terms with the same ___________ should be aligned.
Correct Answer: degree
Question 6:
Subtraction is equivalent to adding the ___________.
Correct Answer: opposite
Question 7:
If the remainder is zero, then the divisor is a ___________ of the dividend.
Correct Answer: factor
Question 8:
The degree of the remainder must be ___________ than the degree of the divisor.
Correct Answer: less
Question 9:
In polynomial long division, you bring down the next term after each ___________ step.
Correct Answer: subtraction
Question 10:
The final answer to a polynomial long division problem is written as the quotient plus the ___________ over the divisor.
Correct Answer: remainder
Educational Standards
Teaching Materials
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