Sum And Difference Of Cubes - Varsity Tutors

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Sum and Difference of Cubes

Study Guide

Key Definition

The sum or difference of two cubes can be factored as $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$ and $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$.

Important Notes

  • Use the mnemonic 'SOAP' for the signs: Same sign, Opposite sign, Always Positive.
  • Factor the Greatest Common Factor (GCF) first if applicable.
  • $x^3 + y^3$ is a sum of cubes.
  • $x^3 - y^3$ is a difference of cubes.
  • Always verify your factorization by expanding.

Mathematical Notation

$+$ represents addition$-$ represents subtraction$\times$ represents multiplication$\div$ represents divisionRemember to use proper notation when solving problems

Why It Works

For example, expanding (x-y)(x^2 + xy + y^2) gives x^3 - y^3: (x-y)(x^2 + xy + y^2) = x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3. Likewise, (x+y)(x^2 - xy + y^2) expands to x^3 + y^3.

Remember

The SOAP mnemonic helps remember the signs during factorization: Same, Opposite, Always Positive.

Quick Reference

Sum of Cubes:$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$Difference of Cubes:$x^3 - y^3 = (x-y)(x^2 + xy + y^2)$

Understanding Sum and Difference of Cubes

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Beginner Explanation

A sum of cubes x^3 + y^3 factors as (x+y)(x^2 - xy + y^2). For example, 8x^3 + 27 = (2x + 3)(4x^2 - 6x + 9).Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What is the factored form of $8x^3 + 27$?

A$(2x + 3)(4x^2 - 6x + 9)$B$(2x - 3)(4x^2 + 6x + 9)$C$(2x + 3)(4x^2 + 6x + 9)$D$(2x - 3)(4x^2 - 6x + 9)$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You have a cubic block of side a and a smaller cube of side b. Express the volume difference between the two cubes using the difference of cubes formula a^3 - b^3 = (a-b)(a^2 + ab + b^2).Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Determine the factors of $64y^3 - 1$ using the difference of cubes.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

What is the factored form of $27p^3 + q^3$?

A$(3p + q)(9p^2 - 3pq + q^2)$B$(3p - q)(9p^2 + 3pq + q^2)$C$(3p + q)(9p^2 + 3pq + q^2)$D$(3p - q)(9p^2 - 3pq + q^2)$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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