Factoring Sums and Differences of Cubes: A PreCalculus Deep Dive

PreAlgebra Grades High School 2:50 Video
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Lesson Description

Master the art of factoring expressions in the form of a^3 + b^3 or a^3 - b^3 using the SOAP mnemonic. This lesson provides a step-by-step guide with examples.

Video Resource

How to factor the Sum of 2 Cubes

Mario's Math Tutoring

Duration: 2:50
Watch on YouTube

Key Concepts

  • Sum of Cubes Formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • Difference of Cubes Formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
  • SOAP mnemonic (Same, Opposite, Always Positive) for remembering the signs in the factored form.

Learning Objectives

  • Students will be able to identify expressions that are sums or differences of perfect cubes.
  • Students will be able to apply the appropriate formula (sum or difference of cubes) to factor the expression correctly.
  • Students will be able to use the SOAP mnemonic to accurately determine the signs in the factored form.
  • Students will be able to verify their factored solutions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of factoring and its importance in simplifying algebraic expressions and solving equations. Briefly touch upon factoring techniques already learned (e.g., factoring out the GCF, difference of squares). Introduce the concept of factoring sums and differences of cubes as an extension of these skills. Show the YouTube video.
  • Formula and SOAP mnemonic (10 mins)
    Present the formulas for factoring the sum and difference of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a^3 - b^3 = (a - b)(a^2 + ab + b^2). Explain the SOAP mnemonic: 'Same' as the original sign, 'Opposite' of the original sign, 'Always Positive'. Emphasize how this helps remember the signs in the factored expression.
  • Example 1: Factoring x^3 + 8 (10 mins)
    Work through the example from the video: x^3 + 8. First, identify 'a' as x and 'b' as 2 (since 8 = 2^3). Apply the sum of cubes formula, paying close attention to the signs. Show the step-by-step process: (x + 2)(x^2 - 2x + 4). Explain each term and its origin in the formula.
  • Example 2: Factoring 27c^3 - 64d^3 (15 mins)
    Work through the second example from the video: 27c^3 - 64d^3. Identify 'a' as 3c and 'b' as 4d (since 27c^3 = (3c)^3 and 64d^3 = (4d)^3). Apply the difference of cubes formula, emphasizing the signs. Show the step-by-step process: (3c - 4d)(9c^2 + 12cd + 16d^2). Have students explain the origin of each term as you write it.
  • Practice Problems (15 mins)
    Provide students with a set of practice problems to work on individually or in pairs. Include problems with varying levels of difficulty. Walk around the classroom to provide assistance and answer questions. Example Problems: 1. m^3 + 1 2. 8p^3 - 125 3. 64x^3 + 27y^3 4. a^3 - 64 5. 216b^3 + 1
  • Review and Wrap-up (5 mins)
    Review the key concepts and steps involved in factoring sums and differences of cubes. Answer any remaining questions. Preview upcoming topics related to polynomial factorization.

Interactive Exercises

  • Whiteboard Challenge
    Divide the class into small groups. Give each group a sum or difference of cubes expression to factor on a whiteboard. The first group to correctly factor the expression wins a small prize.
  • Online Practice
    Direct students to an online resource (e.g., Khan Academy, IXL) with practice problems on factoring sums and differences of cubes. Have them complete a set number of problems and report their scores.

Discussion Questions

  • Why is it important to recognize perfect cubes when factoring?
  • How does the SOAP mnemonic simplify the factoring process?
  • Can all binomials be factored using the sum/difference of cubes formulas? Why or why not?

Skills Developed

  • Algebraic manipulation
  • Pattern recognition
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the factored form of a^3 + b^3?

Correct Answer: (a + b)(a^2 - ab + b^2)

Question 2:

What does the 'O' in the SOAP mnemonic stand for?

Correct Answer: Opposite

Question 3:

Factor x^3 - 27.

Correct Answer: (x - 3)(x^2 + 3x + 9)

Question 4:

Which of the following expressions is a sum of cubes?

Correct Answer: x^3 + 8

Question 5:

What is the factored form of 8a^3 + 1?

Correct Answer: (2a + 1)(4a^2 - 2a + 1)

Question 6:

Factor 64 - y^3

Correct Answer: (4 - y)(16 + 4y + y^2)

Question 7:

What is 'a' in the expression 27x^3 + 8y^3?

Correct Answer: 3x

Question 8:

What is 'b' in the expression x^3 - 125?

Correct Answer: 5

Question 9:

Factor 1 + 64p^3

Correct Answer: (1 + 4p)(1 - 4p + 16p^2)

Question 10:

Factor 8x^3 - 27y^3

Correct Answer: (2x - 3y)(4x^2 + 6xy + 9y^2)

Fill in the Blank Questions

Question 1:

The sum of cubes formula is a^3 + b^3 = (a + b)(a^2 - __ + b^2)

Correct Answer: ab

Question 2:

In the SOAP mnemonic, 'A' stands for ______.

Correct Answer: Always

Question 3:

The factored form of x^3 - 1 is (x - 1)(x^2 + x + __).

Correct Answer: 1

Question 4:

To factor 8x^3 + 27, 'a' is equal to ____.

Correct Answer: 2x

Question 5:

The factored form of 64 - y^3 is (4 - y)(16 + 4y + ____).

Correct Answer: y^2

Question 6:

When factoring a difference of cubes, the first factor will have the ______ sign as the original expression.

Correct Answer: same

Question 7:

The factored form of m^3 + 64 is (m + 4)(m^2 - 4m + ____).

Correct Answer: 16

Question 8:

In the expression 27c^3 - 64d^3, b is equal to ____.

Correct Answer: 4d

Question 9:

The factored form of 1 + 8p^3 is (1 + 2p)(1 - 2p + ____).

Correct Answer: 4p^2

Question 10:

In the SOAP acronym, the O stands for ____.

Correct Answer: opposite

Educational Standards

PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

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