Factoring The Sum Of Cubes: X³ + 216

When you encounter a mathematical expression like $x^3 + 216$, and you're asked to find its factored form, you're likely dealing with a sum of cubes. The sum of cubes is a special algebraic identity that allows us to break down expressions of the form $a^3 + b^3$ into a product of two factors. Recognizing these patterns is a fundamental skill in algebra, and understanding the sum of cubes formula will make factoring much more efficient.

Let's dive into what the sum of cubes formula is. It states that for any two terms, $a$ and $b$, the expression $a^3 + b^3$ can be factored as $(a + b)(a^2 - ab + b^2)$. To apply this formula to our problem, $x^3 + 216$, we first need to identify our 'a' and 'b' terms. It's pretty clear that $a^3$ is $x^3$, which means $a = x$. Now, for $b^3$, we have 216. We need to figure out what number, when cubed, equals 216. If you try out a few small integers, you'll find that $6 imes 6 imes 6 = 36 imes 6 = 216$. So, $b = 6$.

Now that we have $a = x$ and $b = 6$, we can substitute these values into the sum of cubes formula: $(a + b)(a^2 - ab + b^2)$. This gives us $(x + 6)(x^2 - (x)(6) + 6^2)$. Simplifying this expression, we get $(x + 6)(x^2 - 6x + 36)$. This is the factored form of $x^3 + 216$.

Looking at the options provided:

A. (x-6)\\\left(x^2+6 x+36 ight) B. (x+6)\\\left(x^2-6 x+36 ight) C. (x+6)\\\left(x^2-12 x+36 ight) D. (x+6)\\\left(x^2+12 x+36 ight)

We can see that our derived factored form, $(x + 6)(x^2 - 6x + 36)$, matches option B exactly. This confirms that option B is the correct answer.

It's important to remember the difference between the sum of cubes and the difference of cubes. The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Notice the signs are different. The 'a' term is subtracted in the first factor, and the 'ab' term is added in the second factor. For the sum of cubes, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, 'a' is added in the first factor, and the 'ab' term is subtracted in the second factor. Getting these signs right is crucial for accurate factoring. Also, pay close attention to the perfect squares ($a^2$ and $b^2$) in the second factor, as these are often where mistakes can happen, especially if one of the terms is not a simple integer.

Let's consider why the other options are incorrect. Option A has $(x-6)$ as the first factor. If we were factoring a difference of cubes, like $x^3 - 216$, this might be a starting point. However, for a sum of cubes, the first factor must be $(x+6)$. Options C and D have the correct first factor, $(x+6)$, but the second factor is incorrect. In option C, we have $(x^2 - 12x + 36)$. The middle term should be $-ab$, which is $-x(6) = -6x$, not $-12x$. In option D, we have $(x^2 + 12x + 36)$. Again, the middle term is incorrect, and it should also be negative. The last term, $36$, is correct in options A, B, and D because it's $6^2$. In option C, it's also $6^2$, which is correct, but the middle term is wrong.

Mastering factoring techniques like the sum of cubes is essential for simplifying algebraic expressions, solving equations, and understanding more advanced mathematical concepts. Practice with various examples, and you'll find that recognizing these patterns becomes second nature. Remember to always double-check your work by expanding the factored form to see if you get the original expression back. This simple step can save you from many errors. For more resources on algebraic identities and factoring, you can explore Khan Academy's algebra section.