Problem 61 - Algebra and Trigonometry

Question

Factor using the formula for the sum or difference of two cubes. $$8 x^{3}-1$$

Step-by-Step Solution

Answer

The factored form of the equation \(8x^3 - 1\) is \((2x-1)(4x^2 + 2x + 1)\)

Step 1: Identifying a and b

Identify the values of \(a\) and \(b\). In this case, \(a = 2x\) and \(b = 1\) because \(8x^3\) is the cube of \(2x\) and 1 is the cube of 1.

Step 2: Applying the formula

Apply the difference of cubes formula, \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Plug in the values of \(a\) and \(b\) identified in Step 1 into the formula to transform the expression into factored form. The factored form of the equation should look something like this: \((2x-1)((2x)^2 + (2x)(1) + (1)^2)\)

Step 3: Simplifying

Simplify the latter part of the equation obtained in step 2. This results in: \((2x-1)(4x^2 + 2x + 1)\)