Sum & Difference of Cubes: Videos & Practice Problems

A sum or difference of cubes refers to a polynomial expression consisting of two terms that are perfect cubes, either added or subtracted. Recognizing these forms allows us to factor them efficiently using specific formulas derived from their product patterns. For example, expressions like x³ + 8 or 27y³ - 1 fit this category because both terms are perfect cubes.

The factoring rules for sums and differences of cubes are as follows: a sum of cubes, a³ + b³, factors into (a + b)(a² - ab + b²), while a difference of cubes, a³ - b³, factors into (a - b)(a² + ab + b²). These formulas break down the original cubic expression into a product of a binomial and a trinomial, simplifying the polynomial.

To apply these rules, first identify the cube roots of each term. For instance, in x³ + 8, the cube root of x³ is x, and the cube root of 8 is 2, since 2 × 2 × 2 = 8. Using the sum of cubes formula, this factors to (x + 2)(x² - 2x + 4). Similarly, for 27y³ - 1, recognize that 27 is 3³ and 1 is 1³, so the expression becomes (3y)³ - 1³. Applying the difference of cubes formula yields (3y - 1)(9y² + 3y + 1).

To keep track of the signs when factoring sums or differences of cubes, the acronym SOAP is useful: Same, Opposite, Always Positive. This means the sign in the binomial factor matches the original sign between the cubes (same), the sign of the middle term in the trinomial is the opposite, and the last term in the trinomial is always positive. This helps avoid common mistakes in sign placement during factoring.

Understanding how to factor sums and differences of cubes is essential for simplifying complex polynomials and solving higher-degree equations. By identifying perfect cubes and applying the appropriate formula with the SOAP method, factoring becomes a systematic and manageable process.

To factor the polynomial $$\backslash (\backslash frac\{2\}\{3\}\backslash )\; x^3\; -\; \backslash (\backslash frac\{2\}\{3\}\backslash )\; y^3$$, start by identifying the greatest common factor (GCF). Both terms share a factor of $\backslash (\backslash frac\{2\}\{3\}\backslash )$, so factor this out first. This simplifies the expression to $\backslash (\backslash frac\{2\}\{3\}\backslash )\; (x^3\; -\; y^3)$. The remaining expression inside the parentheses is a difference of cubes, which can be factored using the formula for the difference of cubes: $a^3\; -\; b^3\; =\; (a\; -\; b)(a^2\; +\; ab\; +\; b^2)$.

Here, $a\; =\; x$ and $b\; =\; y$, so the factorization becomes:

$$\backslash (\backslash frac\{2\}\{3\}\backslash )\; (x\; -\; y)(x^2\; +\; xy\; +\; y^2)$$

This method highlights the importance of first extracting the GCF to simplify the polynomial before applying the difference of cubes formula. Recognizing the structure of cubic terms and correctly applying the factorization formula ensures accurate and efficient simplification of expressions involving cubes.

To factor the polynomial \(16x^3 + 54y^3\), start by identifying the greatest common factor (GCF). The coefficients 16 and 54 share a GCF of 2. Factoring out 2 gives:

\[ 2(8x^3 + 27y^3) \]

Now, observe that \(8x^3\) and \(27y^3\) are perfect cubes since \(8 = 2^3\) and \(27 = 3^3\). This allows us to recognize the expression inside the parentheses as a sum of cubes, which can be factored using the sum of cubes formula:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

Here, \(a = 2x\) and \(b = 3y\). Applying the formula, we get:

\[ (2x + 3y)( (2x)^2 - (2x)(3y) + (3y)^2 ) \]

Simplifying the terms inside the second parentheses:

\[ (2x + 3y)(4x^2 - 6xy + 9y^2) \]

Therefore, the fully factored form of the original polynomial is:

\[ 2(2x + 3y)(4x^2 - 6xy + 9y^2) \]

This process highlights the importance of first extracting the greatest common factor before applying special factoring formulas such as the sum of cubes. Recognizing perfect cubes and correctly applying the sum of cubes identity enables efficient polynomial factorization.