Sum and Difference of Cubes
Learn to factor expressions in the form a³ + b³ and a³ - b³ using special formulas.
Definition
Sum of Cubes Formula
Difference of Cubes Formula
- Same sign (as the original)
- Opposite sign
- Always Positive
- First factor: same sign as original
- Middle term of trinomial: opposite sign
- Last term of trinomial: always positive
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Worked Examples
Factor
Identify the perfect cubes
and → ,
Apply the sum of cubes formula
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Verify by expanding (optional)
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Answer:
Common Mistakes
Confusing the sign pattern in the trinomial
Why it's wrong: Students often forget whether the middle term is positive or negative.
Correct: Use SOAP: Same, Opposite, Always Positive. The middle term has the OPPOSITE sign from the binomial factor.
Thinking can be factored further
Why it's wrong: This trinomial looks like it might factor, but it cannot be factored over the real numbers.
Correct: The trinomials and are both prime (cannot be factored further).
Forgetting to identify cubes correctly
Why it's wrong: Students may not recognize numbers like , , or as perfect cubes.
Correct: Memorize the first ten cubes: .
Not factoring out the GCF first
Why it's wrong: Jumping straight to cube formulas without checking for common factors.
Correct: Always check for a GCF before applying special factoring formulas.
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Practice Problems
18 problemsWhich of the following is a perfect cube?
Why It Matters
- Simplifying algebraic expressions: Many complex expressions can be factored using these formulas
- Solving cubic equations: Recognizing cube patterns helps solve equations like
- Calculus: These patterns appear when simplifying limits and derivatives
- Physics and engineering: Volume calculations often involve cubic expressions
Real World Applications
Volume Calculations
When calculating the difference between two cubic containers, sum and difference of cubes formulas simplify the algebra.
Example:
A large cube has side length and a small cube has side length . The difference in volumes is .
A sculptor removes a cube with 3 cm sides from a cube with 5 cm sides.
Factor the expression for the remaining volume:
Step 1: Write the mathematical expression
Use the difference of cubes formula with and :
Engineering and Physics
Cubic relationships appear in formulas for power, energy, and fluid dynamics.
Example:
The kinetic energy of wind is proportional to the cube of wind speed. Comparing two wind speeds involves cube expressions.
Wind power at speed is proportional to . How much more power does a 10 m/s wind have compared to a 4 m/s wind?
Simplify by factoring
Step 1: Write the mathematical expression
Apply difference of cubes:
Key Takeaways
- 1Sum of cubes:
- 2Difference of cubes:
- 3Use SOAP to remember signs: Same, Opposite, Always Positive
- 4Perfect cubes to memorize:
- 5The trinomial factor () cannot be factored further over real numbers
- 6Always factor out the GCF before applying cube formulas
Frequently Asked Questions
Glossary
- Perfect cube
- A number that can be written as where is an integer (e.g., , )
- Sum of cubes
- An expression in the form , which factors as
- Difference of cubes
- An expression in the form , which factors as
- SOAP
- Memory device: Same, Opposite, Always Positive - describes the signs in cube factoring formulas
Formula Card
Sum of Cubes
Factor the sum of two cubes into a binomial and trinomial
Difference of Cubes
Factor the difference of two cubes into a binomial and trinomial
SOAP Pattern
Same, Opposite, Always Positive - sign pattern mnemonic