Unlocking the Secrets of Sum and Difference of Cubes

Algebra 2 Grades High School 5:35 Video
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Lesson Description

Master factoring sum and difference of cubes with this comprehensive lesson. Learn the formulas, mnemonic device (SOAP), and work through examples to solidify your understanding.

Video Resource

Factoring Sum and Difference of Cubes

Mario's Math Tutoring

Duration: 5:35
Watch on YouTube

Key Concepts

  • Perfect Cubes
  • Sum of Cubes Formula
  • Difference of Cubes Formula
  • SOAP mnemonic (Same, Opposite, Always Positive)

Learning Objectives

  • Identify perfect cubes and recognize expressions that are sums or differences of cubes.
  • Apply the sum and difference of cubes formulas to factor polynomial expressions.
  • Utilize the SOAP mnemonic to correctly apply the signs in the factored form.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing perfect cubes (1, 8, 27, 64, 125, etc.) and their importance in recognizing sum and difference of cubes patterns. Briefly discuss the formulas for sum and difference of cubes.
  • Formula and SOAP Mnemonic (10 mins)
    Introduce the formulas for factoring the sum and difference of cubes: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²). Explain the SOAP mnemonic (Same, Opposite, Always Positive) to help students remember the signs in the formulas. Emphasize that 'a' and 'b' represent the cube roots of the terms.
  • Example Problems (20 mins)
    Work through the example problems from the video, demonstrating how to identify 'a' and 'b', apply the correct formula, and use the SOAP mnemonic to determine the signs. Explain each step clearly. Emphasize that the resulting trinomial is generally not factorable. Include the following examples: 1. x³ - 27 2. 8y³ + 1 3. 64d³ - 125 4. 216c³ + 1000d³
  • Practice Problems (15 mins)
    Provide students with practice problems to work on individually or in pairs. Circulate to provide assistance and answer questions.
  • Review and Wrap-up (5 mins)
    Review the key concepts and formulas. Answer any remaining questions. Preview the next lesson on factoring techniques.

Interactive Exercises

  • Perfect Cube Identification
    Present students with a list of numbers and ask them to identify which are perfect cubes. This can be done as a quick class activity or a worksheet.
  • Formula Application
    Provide students with expressions that are sums or differences of cubes and have them identify 'a' and 'b', then write out the factored form using the appropriate formula and the SOAP mnemonic.

Discussion Questions

  • What are some common perfect cubes you should memorize?
  • How does the SOAP mnemonic help you remember the formula for factoring sum and difference of cubes?
  • Why is it important to look for a greatest common factor (GCF) before attempting to factor the sum or difference of cubes?

Skills Developed

  • Factoring Polynomials
  • Pattern Recognition
  • Algebraic Manipulation

Multiple Choice Questions

Question 1:

What is the factored form of x³ + 8?

Correct Answer: (x + 2)(x² - 2x + 4)

Question 2:

What does the 'O' in SOAP stand for when factoring sum or difference of cubes?

Correct Answer: Opposite

Question 3:

Which of the following is a perfect cube?

Correct Answer: 64

Question 4:

What is the factored form of 27a³ - 1?

Correct Answer: (3a - 1)(9a² + 3a + 1)

Question 5:

When factoring the sum of cubes, the trinomial factor will always have a _______ last term.

Correct Answer: Positive

Question 6:

Which of the following expressions is a difference of cubes?

Correct Answer: x³ - 8

Question 7:

What is the first step in factoring any polynomial?

Correct Answer: Look for the greatest common factor (GCF)

Question 8:

The factored form of a sum or difference of cubes will contain a binomial and a:

Correct Answer: Trinomial

Question 9:

What is 'a' in the expression x³ + 64 when factoring the sum of cubes?

Correct Answer: x

Question 10:

What is 'b' in the expression 8y³ - 1 when factoring the difference of cubes?

Correct Answer: 1

Fill in the Blank Questions

Question 1:

The acronym ________ helps remember the signs when factoring sum and difference of cubes.

Correct Answer: SOAP

Question 2:

In the sum of cubes formula, a³ + b³ = (a + b)(a² - ab + b²), the sign between 'ab' is ________.

Correct Answer: negative

Question 3:

The cube root of 27 is ________.

Correct Answer: 3

Question 4:

Before attempting to factor a sum or difference of cubes, you should always check for a ________.

Correct Answer: GCF

Question 5:

The formula for the difference of cubes is a³ - b³ = (a - b)(a² + ab + b²). The sign between 'a' and 'b' in the first binomial is the ________ sign as the original expression.

Correct Answer: same

Question 6:

The cube root of 64 is ________.

Correct Answer: 4

Question 7:

The cube root of 125 is ________.

Correct Answer: 5

Question 8:

The last term in the trinomial factor when using the sum or difference of cubes will ________ be positive.

Correct Answer: always

Question 9:

The factored form of x³ - 1 is (x - 1)(x² + x + 1). The sign of the middle term in the trinomial is the ________ sign of the original expression.

Correct Answer: opposite

Question 10:

When factoring x³ + 8, the 'a' term is _______.

Correct Answer: x

Educational Standards

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.

Teaching Materials

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