Factoring a Sum of Cubes

Factor $x^3 + 8$

Identify the perfect cubes: $x^3 = (x)^3$ and $8 = 2^3$ = $a = x$, $b = 2$

Apply the sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ = $(x + 2)(x^2 - 2x + 4)$

Verify by expanding (optional): $(x + 2)(x^2 - 2x + 4) = x^3 - 2x^2 + 4x + 2x^2 - 4x + 8$ = $x^3 + 8$ \checkmark

Answer: $(x + 2)(x^2 - 2x + 4)$

Factoring a Difference of Cubes

Factor $27y^3 - 64$

Identify the perfect cubes: $27y^3 = (3y)^3$ and $64 = 4^3$ = $a = 3y$, $b = 4$

Apply the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ = $(3y - 4)((3y)^2 + (3y)(4) + 4^2)$

Simplify the trinomial: $(3y)^2 = 9y^2$, $(3y)(4) = 12y$, $4^2 = 16$ = $(3y - 4)(9y^2 + 12y + 16)$

Answer: $(3y - 4)(9y^2 + 12y + 16)$

Factoring with a Common Factor First

Factor $2x^3 + 54$

Factor out the GCF: $2x^3 + 54 = 2(x^3 + 27)$ = $2(x^3 + 27)$

Identify the cubes inside: $x^3 = (x)^3$ and $27 = 3^3$ = $a = x$, $b = 3$

Apply sum of cubes formula: $x^3 + 27 = (x + 3)(x^2 - 3x + 9)$ = $(x + 3)(x^2 - 3x + 9)$

Write the complete factorization: Include the GCF from step 1 = $2(x + 3)(x^2 - 3x + 9)$

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

Factoring with Variable Coefficients

Factor $8a^3 - 125b^3$

Identify the perfect cubes: $8a^3 = (2a)^3$ and $125b^3 = (5b)^3$ = First cube: $2a$, Second cube: $5b$

Apply difference of cubes formula: $(2a)^3 - (5b)^3 = (2a - 5b)((2a)^2 + (2a)(5b) + (5b)^2)$ = $(2a - 5b)(4a^2 + 10ab + 25b^2)$

Verify the trinomial coefficients: $(2a)^2 = 4a^2$, $(2a)(5b) = 10ab$, $(5b)^2 = 25b^2$ = All terms are correct

Answer: $(2a - 5b)(4a^2 + 10ab + 25b^2)$

Mistake: Confusing the sign pattern in the trinomial

Why: 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.

Mistake: Thinking $a^2 - ab + b^2$ can be factored further

Why: This trinomial looks like it might factor, but it cannot be factored over the real numbers.

Correct: The trinomials $a^2 - ab + b^2$ and $a^2 + ab + b^2$ are both prime (cannot be factored further).

Mistake: Forgetting to identify cubes correctly

Why: Students may not recognize numbers like $27$, $64$, or $125$ as perfect cubes.

Correct: Memorize the first ten cubes: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$.

Mistake: Not factoring out the GCF first

Why: Jumping straight to cube formulas without checking for common factors.

Correct: Always check for a GCF before applying special factoring formulas.

Volume Calculations

When calculating the difference between two cubic containers, sum and difference of cubes formulas simplify the algebra.

A large cube has side length $x + 2$ and a small cube has side length $x$. The difference in volumes is $(x+2)^3 - x^3$.

Engineering and Physics

Cubic relationships appear in formulas for power, energy, and fluid dynamics.

The power a wind turbine can take from the wind is proportional to the cube of the wind speed. Comparing two wind speeds leads to cube expressions.

Q: Why can't we factor the trinomial further?

A: Read $a^2 - ab + b^2$ as a quadratic in $a$. Its discriminant is $(-b)^2 - 4(1)(b^2) = -3b^2$, which is negative for every $b \neq 0$. So it has no real roots and cannot be factored over the real numbers. The same works for $a^2 + ab + b^2$.

Q: How do I know if a number is a perfect cube?

A: Check if you can find an integer that, when cubed, equals the number. For example, $64 = 4^3$ because $4 \times 4 \times 4 = 64$. Memorize common cubes: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$.

Q: What's the difference between difference of squares and difference of cubes?

A: Difference of squares: $a^2 - b^2 = (a-b)(a+b)$ gives two binomials. Difference of cubes: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$ gives a binomial and a trinomial.