%2F(x-5).jpeg "(x^3-125)/(x-5)")
Simplifying the Expression (x^3 - 125) / (x - 5)
This expression represents a rational function, a fraction where both the numerator and denominator are polynomials. We can simplify this expression by using the difference of cubes factorization.
Understanding the Difference of Cubes
The difference of cubes factorization states that: a³ - b³ = (a - b)(a² + ab + b²)
In our expression, we have:
- a³ = x³
- b³ = 125 = 5³
Applying the Factorization
Let's apply the difference of cubes factorization to the numerator:
(x³ - 125) = (x - 5)(x² + 5x + 25)
Now our expression becomes:
[(x - 5)(x² + 5x + 25)] / (x - 5)
Cancellation and Final Result
Notice that we have a common factor of (x - 5) in both the numerator and denominator. We can cancel this factor, leaving us with:
x² + 5x + 25
Therefore, the simplified form of the expression (x³ - 125) / (x - 5) is x² + 5x + 25.
Important Note
It's important to remember that this simplification is valid only when x ≠ 5. If x = 5, the denominator becomes zero, making the expression undefined.