- Recognize \(x^3 + 216\) as a sum of cubes: \(x^3 + 6^3\).
- Use the identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) with \(a=x\) and \(b=6\).
- Substitute the variables to get \((x+6)(x^2 - 6x + 36)\).
- Final answer: \(\boxed{(x + 6)(x^2 - 6x + 36)}\).
Explanation
- Identify the expression structure
The expression \(x^3 + 216\) can be identified as a sum of two cubes because 216 is a perfect cube: \(6^3 = 6 \times 6 \times 6 = 216\). Thus, the expression can be rewritten as \(x^3 + 6^3\). We can use the sum of cubes factorization formula, which states that for any two numbers \(a\) and \(b\), \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
- Apply the factorization formula
Applying the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) with \(a = x\) and \(b = 6\), we perform the following substitutions:
- \(a + b = x + 6\)
- \(a^2 = x^2\)
- \(ab = x \times 6 = 6x\)
- \(b^2 = 6^2 = 36\)
Substituting these values into the formula gives:
\(x^3 + 6^3 = (x + 6)(x^2 - 6x + 36)\).
- Select the correct option
Comparing our result \((x + 6)(x^2 - 6x + 36)\) with the provided options:
- \((x - 6)(x^2 + 6x + 36)\)
- \((x + 6)(x^2 - 6x + 36)\)
- \((x + 6)(x^2 - 12x + 36)\)
- \((x + 6)(x^2 + 12x + 36)\)
The second option perfectly matches our calculated factored form. Therefore, the correct answer is \((x + 6)(x^2 - 6x + 36)\).
Examples
Factoring expressions like the sum of cubes is fundamental in algebra for simplifying complex polynomial fractions and solving higher-degree equations. For instance, if you were analyzing the motion of an object where the position is defined by a cubic function like \(P(t) = t^3 + 216\), being able to factor it into \((t + 6)(t^2 - 6t + 36)\) allows you to easily identify roots, which might correspond to specific time intervals when the object is at a particular position or speed. This technique is widely used in engineering and physics to solve for variables in non-linear models.