Problem 64 - Algebra and Trigonometry

Question

Factor using the formula for the sum or difference of two cubes. $$8 x^{3}+125$$

Step-by-Step Solution

Answer

The factorized form of the given expression is \( (2x + 5)(4x^{2} - 10x + 25)\)

Step 1: Identify the Cubes

First, rewrite \(8x^{3}+125\) as \( (2x)^{3} + (5)^{3} \) to clearly identify the cubes in the expression.

Step 2: Apply the Sum of Cubes Formula

Using the sum of cubes factoring formula, \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\), where \(a = (2x)\) and \(b = 5\). Substitute \(a\) and \(b\) into the formula.

Step 3: Simplify

After putting \(a\) and \(b\) into the formula, simplify the expression. This gives, \((2x)^{3} + 5^{3} = (2x + 5)((2x)^{2} - (2x)(5) + 5^{2}) = (2x + 5)(4x^{2} - 10x + 25)\)