Unlocking Polynomial Secrets: Synthetic Division and Factorization

PreAlgebra Grades High School 2:58 Video
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Lesson Description

Master synthetic division to find remaining factors of polynomials, determine zeros, write complete linear factorizations, and sketch polynomial graphs.

Video Resource

Find Remaining Factors When Given 1 Factor (Synthetic Division)

Mario's Math Tutoring

Duration: 2:58
Watch on YouTube

Key Concepts

  • Synthetic Division
  • Factor Theorem
  • Zeros of a Polynomial
  • Complete Linear Factorization
  • Sketching Polynomial Graphs

Learning Objectives

  • Use synthetic division to find remaining factors when given one factor of a polynomial.
  • Determine the zeros of a polynomial function.
  • Write the complete linear factorization of a polynomial.
  • Sketch a basic graph of a polynomial function using its zeros and end behavior.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the Factor Theorem and the relationship between factors, zeros, and x-intercepts of a polynomial. Introduce the concept of synthetic division as a tool for finding remaining factors.
  • Video Viewing (10 mins)
    Watch the Mario's Math Tutoring video 'Find Remaining Factors When Given 1 Factor (Synthetic Division)'. Encourage students to take notes on the steps of synthetic division and how to interpret the results.
  • Synthetic Division Practice (15 mins)
    Work through examples of synthetic division together, emphasizing the importance of placeholders for missing terms. Start with simpler examples and gradually increase complexity.
  • Finding Zeros and Factorization (15 mins)
    Demonstrate how to find the zeros from the factors obtained after synthetic division. Explain how to write the complete linear factorization of the polynomial. Provide examples.
  • Graph Sketching (10 mins)
    Review how to use the zeros and leading coefficient to sketch a basic graph of the polynomial. Discuss end behavior and the effect of repeated roots (if applicable).

Interactive Exercises

  • Synthetic Division Challenge
    Provide students with polynomials and a given factor. Have them use synthetic division to find the remaining factors and zeros.
  • Factorization Frenzy
    Give students polynomials. Ask them to use synthetic division and factoring to rewrite each polynomial as a product of linear factors and then state all zeros of the polynomial.
  • Graphing Gallery
    Provide students with polynomials. Ask them to graph polynomials after finding the zeros and linear factorization.

Discussion Questions

  • How does the Factor Theorem relate to synthetic division?
  • What does a zero remainder in synthetic division tell you?
  • How do you determine the end behavior of a polynomial function?
  • Why is it important to use placeholders in synthetic division?

Skills Developed

  • Synthetic Division
  • Polynomial Factorization
  • Finding Zeros of Polynomials
  • Graphing Polynomials

Multiple Choice Questions

Question 1:

What is the first step in performing synthetic division?

Correct Answer: Drop down the leading coefficient.

Question 2:

If you are given a factor of (x - 3), what value do you use in the 'box' for synthetic division?

Correct Answer: 3

Question 3:

What does a zero remainder in synthetic division indicate?

Correct Answer: The divisor is a factor.

Question 4:

After performing synthetic division on a cubic polynomial, the quotient will be a polynomial of what degree?

Correct Answer: Quadratic

Question 5:

If the zeros of a polynomial are -2, 1, and 3, which of the following is the complete linear factorization?

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

Question 6:

Which of the following is NOT a factor of x³ - x² - 5x - 3, given that (x-3) is a factor?

Correct Answer: x - 1

Question 7:

What is the constant term in synthetic division called?

Correct Answer: Remainder

Question 8:

When sketching a polynomial graph, the zeros represent the:

Correct Answer: X-intercepts

Question 9:

If synthetic division results in a quotient of x² + 2x - 1, what are you left with?

Correct Answer: A quadratic function

Question 10:

What should you do if a term is missing when setting up synthetic division (e.g., going from x³ to x)?

Correct Answer: Use a placeholder of 0.

Fill in the Blank Questions

Question 1:

The Factor Theorem states that if f(a) = 0, then (x - ___) is a factor of f(x).

Correct Answer: a

Question 2:

In synthetic division, we use the ___ sign of the constant term from the given factor.

Correct Answer: opposite

Question 3:

The last number in synthetic division is the ___.

Correct Answer: remainder

Question 4:

If (x + 5) is a factor, then x = ___ is a zero of the polynomial.

Correct Answer: -5

Question 5:

The process of writing a polynomial as a product of linear factors is called complete linear ___.

Correct Answer: factorization

Question 6:

Before using synthetic division, the polynomial should be written in ___ order.

Correct Answer: descending

Question 7:

The x-intercepts of a polynomial are also known as its ___.

Correct Answer: zeros

Question 8:

To sketch a polynomial graph, you need to know the zeros and the ___ behavior.

Correct Answer: end

Question 9:

When a term is missing in the polynomial, add a zero as a ___.

Correct Answer: placeholder

Question 10:

A remainder of zero after synthetic division tells us that our divisor is a ___ of the polynomial.

Correct Answer: factor

Educational Standards

PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.2:Graphically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features.

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