Unlocking Polynomial Secrets: Synthetic Division and the Division Algorithm

Algebra 2 Grades High School 2:38 Video
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Lesson Description

Master synthetic division and express polynomials in the form P(x) = D(x)Q(x) + r. Learn to relate polynomial division to numerical division and apply this knowledge to solve algebraic problems.

Video Resource

Synthetic Division and P(x)=D(x)Q(x) + r

Mario's Math Tutoring

Duration: 2:38
Watch on YouTube

Key Concepts

  • Synthetic Division
  • Polynomial Division Algorithm: P(x) = D(x)Q(x) + r
  • Divisor, Quotient, Remainder

Learning Objectives

  • Apply synthetic division to divide polynomials.
  • Express a polynomial as the product of the divisor and quotient, plus the remainder.
  • Relate polynomial division to numerical division.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing long division with numbers to build an analogy to polynomial division. Watch the first 20 seconds of the video to reinforce this concept. Discuss the goal of expressing P(x) as D(x)Q(x) + r.
  • Synthetic Division Explained (10 mins)
    Explain the process of synthetic division using the example from the video (1:06-1:57). Emphasize the importance of placeholders for missing terms (e.g., 0x^3). Guide students through the steps: writing the coefficients, bringing down the first coefficient, multiplying, and adding.
  • Writing P(x) = D(x)Q(x) + r (5 mins)
    Show how to write the final answer in the form P(x) = D(x)Q(x) + r (1:57-end). Explain how the coefficients obtained from synthetic division form the quotient, and how the last number represents the remainder. Stress that the degree of the quotient is one less than the degree of the original polynomial.
  • Practice Problems (15 mins)
    Provide students with practice problems to work on individually or in pairs. Example: (2x^3 - 5x + 3) divided by (x + 1). Have students check their answers with each other or with a provided solution key.
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts covered in the lesson. Answer any remaining questions students may have. Preview the next lesson on polynomial equations.

Interactive Exercises

  • Online Synthetic Division Calculator
    Use an online synthetic division calculator to check answers and explore different problems. Students can input the polynomial and divisor and compare their work with the calculator's output.
  • Group Problem Solving
    Divide the class into small groups and assign each group a different polynomial division problem. Have each group present their solution to the class, explaining each step of the synthetic division process and how they arrived at the final answer in the form P(x) = D(x)Q(x) + r.

Discussion Questions

  • How is polynomial division similar to numerical division?
  • Why is it important to include placeholders (zeros) for missing terms when using synthetic division?
  • How does the degree of the quotient relate to the degree of the original polynomial?

Skills Developed

  • Algebraic Manipulation
  • Problem Solving
  • Critical Thinking

Multiple Choice Questions

Question 1:

What is the general form for expressing a polynomial P(x) after division by D(x)?

Correct Answer: P(x) = D(x)Q(x) + r

Question 2:

When using synthetic division, what should you do if a term is missing (e.g., no x^2 term)?

Correct Answer: Insert a placeholder of 0 for the coefficient.

Question 3:

In synthetic division, the last number obtained represents the:

Correct Answer: Remainder

Question 4:

If you divide a polynomial of degree 4 by a linear factor (degree 1) using synthetic division, what is the degree of the resulting quotient?

Correct Answer: 3

Question 5:

When dividing by (x - 3) using synthetic division, what value is placed in the 'box'?

Correct Answer: 3

Question 6:

What is D(x) in the equation P(x) = D(x)Q(x) + r?

Correct Answer: The Divisor

Question 7:

What is the degree of the quotient if you divide x^5 + 2x^3 - x + 5 by x - 1?

Correct Answer: 3

Question 8:

If the remainder is 0 after synthetic division, what does this indicate?

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

Question 9:

Which of the following is the correct setup for dividing (x^3 - 4x + 1) by (x - 2) using synthetic division?

Correct Answer: 2 | 1 0 -4 1

Question 10:

After performing synthetic division, if the quotient is 2x^2 + 3x - 1 and the remainder is 5, which of the following represents the correct expression for P(x) given the divisor (x+1)?

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

Fill in the Blank Questions

Question 1:

In the equation P(x) = D(x)Q(x) + r, Q(x) represents the ______.

Correct Answer: quotient

Question 2:

When using synthetic division, the value used in the division 'box' for (x + 5) is ______.

Correct Answer: -5

Question 3:

A placeholder of ______ must be used when a term is missing in the polynomial during synthetic division.

Correct Answer: 0

Question 4:

The degree of the quotient is always ______ than the degree of the original polynomial when dividing by a linear factor.

Correct Answer: less

Question 5:

The last number calculated in synthetic division represents the ______.

Correct Answer: remainder

Question 6:

If P(x) = (x - 2)(x^2 + 1) + 3, then D(x) is equal to _______.

Correct Answer: x-2

Question 7:

If the remainder is zero, then the divisor is a ______ of the polynomial.

Correct Answer: factor

Question 8:

When dividing (x^3 + 2x - 5) by (x+1) using synthetic division, the first row will start with 1, ______, 2, -5.

Correct Answer: 0

Question 9:

In P(x) = D(x)Q(x) + r, if r is 0, then P(x) = D(x) * _______.

Correct Answer: Q(x)

Question 10:

If synthetic division results in a remainder of -2 when dividing by (x-3), then P(3) = _______.

Correct Answer: -2

Educational Standards

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). A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

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