Question

difference of cubes
8x³ + 25

Explanation:

Step1: Recall the sum of cubes formula

The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). We need to express \(8x^3+25\) in terms of cubes. First, rewrite \(8x^3\) as \((2x)^3\) since \(2x\times2x\times2x = 8x^3\). Now, check if \(25\) is a perfect cube. But \(25 = 5^2\), not a perfect cube. Wait, maybe there's a typo, and it's \(8x^3+125\) (since \(125 = 5^3\)). Assuming it's \(8x^3 + 125\) (as \(25\) is not a cube, but \(125\) is), then \(a = 2x\) and \(b = 5\) because \((2x)^3=8x^3\) and \(5^3 = 125\).

Step2: Apply the sum of cubes formula

Using the formula \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\) with \(a = 2x\) and \(b = 5\), we substitute:
\(a^3 + b^3=(2x)^3+5^3=(2x + 5)((2x)^2-(2x)(5)+5^2)\)
Simplify each part:
\((2x)^2=4x^2\), \((2x)(5) = 10x\), and \(5^2 = 25\). So the factored form is \((2x + 5)(4x^2-10x + 25)\).

(Note: If the original problem is indeed \(8x^3 + 25\), since \(25\) is not a perfect cube, it cannot be factored using the sum or difference of cubes formula. But given the title "Difference of Cubes" (maybe a mislabel, and it's sum), and \(25\) is likely a typo for \(125\), we proceed with the common case.)

Answer:

If the expression is \(8x^3 + 125\) (correcting \(25\) to \(125\)), the factored form is \(\boldsymbol{(2x + 5)(4x^2-10x + 25)}\). If it's \(8x^3+25\), it cannot be factored as a sum/difference of cubes.