Sum of Cubes Calculator
Factor expressions in the form a³ + b³ using the SOAP method. Get complete step-by-step solutions with the formula (a + b)(a² - ab + b²).
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Enter a and b for: (ax)³ + b³
SOAP Mnemonic:
Understanding Sum of Cubes
The sum of cubes is a special factoring pattern used when you have two perfect cubes added together. Unlike the sum of squares (which doesn't factor), the sum of cubes has a specific factoring formula.
The Formula:
a³ + b³ = (a + b)(a² - ab + b²)
The SOAP Method:
Use the SOAP mnemonic to remember the pattern:
- Same: The binomial has the SAME sign as the original (plus)
- Opposite: The trinomial starts with the OPPOSITE sign (minus)
- Always: The middle term is ALWAYS subtraction
- Positive: The last term is always POSITIVE
Common Examples:
x³ + 8 = (x + 2)(x² - 2x + 4)
x³ + 27 = (x + 3)(x² - 3x + 9)
8x³ + 1 = (2x + 1)(4x² - 2x + 1)
x³ + 64 = (x + 4)(x² - 4x + 16)
27x³ + 8 = (3x + 2)(9x² - 6x + 4)
Recognition Checklist:
- Two terms only (binomial)
- Both terms are perfect cubes
- Terms are separated by addition (plus sign)
- Can involve variables and/or constants
Common Mistakes to Avoid:
- Confusing sum of cubes with sum of squares (which doesn't factor)
- Using the wrong signs in the trinomial
- Forgetting that the trinomial usually doesn't factor further
- Making errors when calculating a², ab, or b²
- Not checking if terms are actually perfect cubes
Frequently Asked Questions
What is the sum of cubes formula?
The sum of cubes formula is a³ + b³ = (a + b)(a² - ab + b²). It's used to factor binomials where two perfect cubes are added together. Remember the SOAP mnemonic to keep the signs straight.
What does SOAP stand for in factoring?
SOAP is a mnemonic device: Same (sign as original), Opposite (sign in trinomial first term), Always (middle term is always subtraction), Positive (last term is always positive). This helps you remember the pattern (a + b)(a² - ab + b²).
How do I identify a sum of cubes?
Look for: (1) exactly two terms, (2) both terms are perfect cubes, and (3) the terms are separated by addition. Perfect cubes include 1, 8, 27, 64, 125, x³, 8x³, 27x³, etc.
Can the trinomial factor be factored further?
In most cases, no. The trinomial (a² - ab + b²) resulting from sum of cubes factoring is usually prime and cannot be factored further over real numbers. This is different from some other factoring methods.
What's the difference between sum of cubes and difference of cubes?
The main difference is in the signs. Sum of cubes (a³ + b³) factors as (a + b)(a² - ab + b²), while difference of cubes (a³ - b³) factors as (a - b)(a² + ab + b²). Notice the sign changes in both the binomial and trinomial.
What are some perfect cubes I should memorize?
Memorize these perfect cubes: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729, 10³=1000. Also remember that (2x)³=8x³, (3x)³=27x³, etc.
How do I verify my factoring is correct?
Multiply the factors back together. Multiply the binomial by each term of the trinomial using the distributive property. If you get the original expression (a³ + b³), your factoring is correct.
Why doesn't sum of squares factor but sum of cubes does?
This is due to algebraic properties. While a² + b² is prime over real numbers, a³ + b³ has a factorization that works. The formula (a + b)(a² - ab + b²) always equals a³ + b³ when multiplied out, but no similar simple formula exists for a² + b².
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