When you encounter a binomial expression that doesn’t fit the pattern of a difference of squares, you may need to consider whether it’s a **difference** or **sum of cubes**. Recognizing and factoring these correctly is essential for simplifying polynomials.

### 1. **Difference of Cubes**

A difference of cubes is of the form $a_{3}−b_{3}$. This is factored using the following formula: $a_{3}−b_{3}=(a−b)(a_{2}+ab+b_{2})$

### 2. **Sum of Cubes**

A sum of cubes follows the form $a_{3}+b_{3}$. To factor it, you use: $a_{3}+b_{3}=(a+b)(a_{2}−ab+b_{2})$

**Steps for Factoring a Cubic Difference or Sum**

Let’s use the example $8x_{3}+27$ to walk through the process:

**Step 1: Look for a Greatest Common Factor (GCF)**

Before doing anything else, always check if the terms in the binomial share a common factor. In this case, $8x_{3}$ and $27$ don’t have a GCF, so we can move on to the next step.

**Step 2: Check if it’s a Difference of Squares**

You might think this binomial could be factored as a difference of squares, but the plus sign between the terms immediately rules this out.

**Step 3: Determine if It’s a Sum or Difference of Cubes**

The presence of a plus sign suggests that it may be a sum of cubes. Now, rewrite the terms as cubes: $8x_{3}+27=(2x_{3}+(3_{3}$

Since both terms are cubes, it confirms that this is a sum of cubes.

**Step 4: Apply the Factoring Formula**

Using the sum of cubes formula $a_{3}+b_{3}=(a+b)(a_{2}−ab+b_{2})$, replace $a$ with $2x$ and $b$ with $3$:

$(2x+3)((2x_{2}−(2x)(3)+(3_{2})$

**Step 5: Simplify**

Now simplify the expression:

$(2x+3)(4x_{2}−6x+9)$

**Step 6: Check for Further Factorization**

Once you have factored the expression, always check if it can be simplified further. In this case:

- The binomial term $2x+3$ won’t factor any further since it’s a first-degree binomial.
- The trinomial $4x_{2}−6x+9$ doesn’t factor any further either.

Thus, your final factored form is:

$(2x+3)(4x_{2}−6x+9)$

**Frequently Asked Questions (FAQs)**

**What’s the difference between a difference of cubes and a sum of cubes?**

- A
**difference of cubes**has the form $a_{3}−b_{3}$ and factors as $(a−b)(a_{2}+ab+b_{2})$. - A
**sum of cubes**has the form $a_{3}+b_{3}$ and factors as $(a+b)(a_{2}−ab+b_{2})$.

**How do I recognize a cubic expression?**

A cubic expression will have terms where the variables are raised to the power of three, such as $x_{3}$, $8x_{3}$, or $27$ (which is $_{3}$).

**Can I use the same factoring formula for both sum and difference of cubes?**

No. The formulas are slightly different:

**Difference of cubes**: $a_{3}−b_{3}=(a−b)(a_{2}+ab+b_{2})$**Sum of cubes**: $a_{3}+b_{3}=(a+b)(a_{2}−ab+b_{2})$

**What happens if I forget to check for a GCF before factoring?**

If you skip this step, you might end up with a factored form that isn’t fully simplified. Always check for a GCF first to ensure your final answer is as simple as possible.

**How do I know when I’m done factoring?**

After applying the sum or difference of cubes formula, check the resulting factors:

- If the binomial is first-degree (like $2x+3$) and has no GCF, it’s already in its simplest form.
- For the trinomial, check if it fits other common factoring techniques (such as the difference of squares or quadratic formulas). If it doesn’t, you’re done.

**Can a sum of cubes ever be factored further?**

Generally, after applying the sum of cubes formula, the result is in its simplest form. However, always check for a possible GCF or other factoring methods just to be sure.