Conquering Cubics: Factoring and Solving Cubic Equations

Algebra 2 Grades High School 8:16 Video
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Lesson Description

Learn how to factor and solve cubic equations using the rational root theorem and synthetic division. This lesson builds upon factoring skills and introduces a systematic approach to finding solutions for cubic polynomials.

Video Resource

Solve Cubic Equation Algebra

Kevinmathscience

Duration: 8:16
Watch on YouTube

Key Concepts

  • Rational Root Theorem (Guessing the First Root)
  • Synthetic Division
  • Factoring Trinomials
  • Solving Polynomial Equations

Learning Objectives

  • Students will be able to identify potential rational roots of a cubic equation using the factors of the constant term.
  • Students will be able to use synthetic division to divide a cubic polynomial by a linear factor.
  • Students will be able to factor the resulting quadratic expression after synthetic division.
  • Students will be able to find all real solutions to a cubic equation.

Educator Instructions

  • Introduction (5 mins)
    Briefly review factoring techniques for quadratic equations. Introduce the concept of cubic equations and the challenge of solving them. Explain that this lesson will introduce a method using 'guessing' and synthetic division.
  • Video Viewing and Note-Taking (15 mins)
    Play the Kevinmathscience video "Solve Cubic Equation Algebra". Instruct students to take detailed notes on the steps involved in the process: finding potential roots, using synthetic division, and factoring the resulting quadratic.
  • Worked Examples and Practice (20 mins)
    Work through the examples provided in the video, pausing to clarify any confusing steps. Provide additional examples for students to work through independently or in pairs. Focus on identifying the factors of the constant term and correctly applying synthetic division.
  • Discussion and Q&A (10 mins)
    Facilitate a class discussion to address any remaining questions or misconceptions. Ask students to explain the process in their own words. Discuss the limitations of this method (it only finds rational roots).

Interactive Exercises

  • Root Guessing Game
    Present students with a list of cubic equations. In pairs, they should identify the potential rational roots for each equation. Discuss the strategies they used.
  • Synthetic Division Relay
    Divide the class into teams. Each team member performs one step of synthetic division on a given problem, passing the result to the next team member. The first team to correctly complete the division wins.

Discussion Questions

  • Why do we start by finding the factors of the constant term?
  • What does a remainder of zero in synthetic division tell us?
  • How is factoring the quadratic expression after synthetic division related to finding the remaining roots of the cubic equation?
  • What are some situations where this method might not work well?

Skills Developed

  • Factoring
  • Synthetic Division
  • Problem-Solving
  • Critical Thinking
  • Application of the Rational Root Theorem

Multiple Choice Questions

Question 1:

What is the first step in solving a cubic equation using the method taught in the video?

Correct Answer: Guessing a first root

Question 2:

If -2 is a root of a cubic equation, what does this mean?

Correct Answer: (x + 2) is a factor

Question 3:

What is the purpose of synthetic division in this process?

Correct Answer: To simplify the equation by dividing out a known factor

Question 4:

After performing synthetic division, you are left with a quadratic expression. What should you do next?

Correct Answer: Set it equal to zero and solve

Question 5:

What are the possible rational roots of x³ + 2x² - 5x - 6 = 0?

Correct Answer: ±1, ±2, ±3, ±6

Question 6:

What does a remainder of 0 after synthetic division indicate?

Correct Answer: The number you divided by IS a root

Question 7:

Which of the following is NOT a factor of the constant term in the polynomial x³ - 4x² + x + 6?

Correct Answer: 5

Question 8:

How many real roots can a cubic equation have?

Correct Answer: One, two, or three

Question 9:

If you find one root of a cubic equation, what degree is the polynomial you get after using synthetic division?

Correct Answer: 2

Question 10:

What method should be used to solve for the roots of the simplified polynomial?

Correct Answer: Quadratic Formula/Factoring

Fill in the Blank Questions

Question 1:

The first step in solving a cubic equation by 'guessing' is to find the __________ of the constant term.

Correct Answer: factors

Question 2:

__________ __________ is a method used to divide a polynomial by a linear factor.

Correct Answer: Synthetic division

Question 3:

A remainder of zero after synthetic division indicates that the number you divided by is a __________ of the polynomial.

Correct Answer: root

Question 4:

After synthetic division, the cubic equation is reduced to a __________ equation.

Correct Answer: quadratic

Question 5:

If x = 3 is a solution, then (x - ___) is a factor.

Correct Answer: 3

Question 6:

When using the "guess and check" method, you can test both positive and __________ of each factor.

Correct Answer: negative

Question 7:

The constant term of a polynomial is the term without a _________.

Correct Answer: variable

Question 8:

If a cubic equation has one real root, the other two roots might be __________ numbers.

Correct Answer: imaginary

Question 9:

A cubic equation can have a maximum of _________ real roots.

Correct Answer: three

Question 10:

To find the remaining roots, one can __________ the quotient from synthetic division.

Correct Answer: factor

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.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies.

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