Final answer:
The greatest common factor of 25x⁴ - 10x³ - 5x² is 5x², as it is the largest integer that divides all coefficients and the smallest power of x common to all terms.
Step-by-step explanation:
The student has asked to find the greatest common factor (GCF) of the algebraic expression: 25x⁴ - 10x³ - 5x².
To find the GCF, we need to identify the smallest power of x in all the terms and the largest integer that divides all the coefficients. In this case, the smallest power of x is x² and the largest integer that divides all the coefficients is 5. Thus, the greatest common factor of this expression is 5x².
As an example to understand the process, consider that when two powers of the same base are multiplied, such as 5³ ⋅ 5⁴ (5 to the third power multiplied by 5 to the fourth power), the exponents are added. Therefore, 5³ ⋅ 5⁴ = 5⁷ (5 to the seventh power).
Applying this rule in reverse helps to see why 5x² is the factor being taken out from each term: it's like 'dividing' out the common part from each term to simplify the expression.