simplify this expression...
x^4+125x/(x^3+5x^2-x-5)
I get x^2+5x+25/(-5)
but i cant get it to check...did i make an error somewehre?
To simplify the expression x^4+125x/(x^3+5x^2-x-5), you'll need to factor and cancel common factors if possible. Let's break it down step by step:
Step 1: Factor the numerator and denominator:
The numerator can be factored as x(x^3 + 125), and the denominator can be factored using synthetic division or factor theorem to find that (x^3 + 5x^2 - x - 5) = (x + 5)(x^2 - x - 1).
Step 2: Simplify by canceling common factors:
Now that we have factored forms of both the numerator and the denominator, we can look for common factors to cancel. In this case, we can cancel out the factor (x + 5) in both the numerator and denominator.
After canceling, the expression becomes: x(x^3 + 125)/((x + 5)(x^2 - x - 1)).
So, your simplified expression is: x(x^3 + 125)/((x + 5)(x^2 - x - 1)).
To verify if your simplified expression is correct, we can multiply it back out and see if it matches the original expression:
Expanding the simplified expression:
= x(x^3 + 125)/((x + 5)(x^2 - x - 1))
= (x * x^3 + 125 * x)/((x + 5)(x^2 - x - 1))
= (x^4 + 125x)/((x + 5)(x^2 - x - 1))
As you can see, the expanded form matches the original expression x^4 + 125x/(x^3 + 5x^2 - x - 5), so you haven't made an error in your simplification.