%2F(x%2B5).jpeg "(x^3+125)/(x+5)")
Factoring and Simplifying the Expression (x^3 + 125) / (x + 5)
This expression involves a cubic polynomial in the numerator and a linear term in the denominator. To simplify it, we can use the concept of factoring and canceling out common factors.
Recognizing the Pattern
The numerator, x^3 + 125, is a sum of cubes. This is because 125 is the cube of 5 (5 * 5 * 5 = 125). We can use the following formula to factor a sum of cubes:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In our case, a = x and b = 5. Applying the formula, we get:
x^3 + 125 = (x + 5)(x^2 - 5x + 25)
Simplifying the Expression
Now we can substitute this factored form back into our original expression:
(x^3 + 125) / (x + 5) = [(x + 5)(x^2 - 5x + 25)] / (x + 5)
Notice that we have a common factor of (x + 5) in both the numerator and denominator. We can cancel this out:
[(x + 5)(x^2 - 5x + 25)] / (x + 5) = x^2 - 5x + 25
Result
Therefore, the simplified form of the expression (x^3 + 125) / (x + 5) is x^2 - 5x + 25. This expression is valid for all values of x except for x = -5, where the original expression is undefined.