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 kinetic energy of wind is proportional to the cube of wind speed. Comparing two wind speeds involves cube expressions.

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

A: The discriminant of $a^2 - ab + b^2$ is $b^2 - 4(1)(b^2) = -3b^2 < 0$. Since it's negative, there are no real roots, so it cannot be factored over the real numbers.

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.