Question

1. $8x^3 - 125 div (2x - 5)$

Explanation:

Step1: Recognize the difference of cubes

The numerator \(8x^{3}-125\) can be written as a difference of cubes. Recall that \(a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})\). Here, \(8x^{3}=(2x)^{3}\) and \(125 = 5^{3}\), so \(8x^{3}-125=(2x)^{3}-5^{3}\).
Using the difference of cubes formula, \((2x)^{3}-5^{3}=(2x - 5)((2x)^{2}+(2x)\times5 + 5^{2})=(2x - 5)(4x^{2}+10x + 25)\).

Step2: Perform the division

Now we have \(\frac{(2x - 5)(4x^{2}+10x + 25)}{(2x - 5)}\). Since \(2x-5
eq0\) (to avoid division by zero), we can cancel out the common factor \((2x - 5)\) from the numerator and the denominator.

Answer:

\(4x^{2}+10x + 25\)