Problem 60 - Algebra and Trigonometry

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3}-27$$

Step-by-Step Solution

Answer

The factored form of the expression \(x^3 - 27\) using the formula for the difference of two cubes is \((x - 3)(x^2 + 3x + 9)\).

Step 1: Identify the Cubes

The first step is to identify the cubes in our formula. Here, \(x^3\) is our first cube (\(a^3\)) and \(27\) is our second cube (\(b^3\)). Now we need to figure out the cube roots, which are \(x\) for \(x^3\) and \(3\) for \(27\). Now we have our \(a\) and \(b\), which are \(x\) and \(3\) respectively.

Step 2: Apply the Formula

Now we will apply the formula for the difference of two cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Replacing \(a\) with \(x\) and \(b\) with \(3\), we get: \(x^3 - 3^3 = (x - 3)(x^2 + 3x + (3)^2)\).

Step 3: Simplify the Result

The final step is to simplify the expanded equation: \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\).