To simplify the given rational expression:
\[ \frac{6x}{x+1} - \frac{5}{5x^3} \]
Step 1: Find a common denominator.
The common denominator between \(x+1\) and \(5x^3\) is \(5x^3(x+1)\).
Step 2: Rewrite each fraction with the common denominator.
The first term:
\[ \frac{6x}{x+1} = \frac{6x \cdot 5x^3}{(x+1) \cdot 5x^3} = \frac{30x^4}{5x^3(x+1)} \]
The second term:
\[ \frac{5}{5x^3} = \frac{5 \cdot (x+1)}{5x^3 \cdot (x+1)} = \frac{5(x+1)}{5x^3(x+1)} = \frac{5x + 5}{5x^3(x+1)} \]
Step 3: Combine the fractions.
Now we can combine the fractions:
\[ \frac{30x^4}{5x^3(x+1)} - \frac{5(x+1)}{5x^3(x+1)} = \frac{30x^4 - (5x + 5)}{5x^3(x+1)} \]
Step 4: Simplify the numerator.
Distributing the negative sign in the numerator:
\[ 30x^4 - 5x - 5 \]
So, we have:
\[ \frac{30x^4 - 5x - 5}{5x^3(x+1)} \]
Step 5: Factor the numerator if possible.
Let's factor the numerator \(30x^4 - 5x - 5\):
We can try factoring out \(5\) from the numerator:
\[ 5(6x^4 - x - 1) \]
Thus, the expression now looks like:
\[ \frac{5(6x^4 - x - 1)}{5x^3(x+1)} \]
The \(5\) cancels out:
\[ \frac{6x^4 - x - 1}{x^3(x+1)} \]
Step 6: Final simplified form.
The simplified rational expression is:
\[ \frac{6x^4 - x - 1}{x^3(x+1)} \]
Step 7: Define the constraints.
The expression is undefined when the denominator equals zero. To find constraints:
- \(x^3 = 0 \implies x = 0\)
- \(x + 1 = 0 \implies x = -1\)
Thus, \(x\) cannot equal \(0\) or \(-1\).
Final answer:
The simplified expression is:
\[ \frac{6x^4 - x - 1}{x^3(x+1)} \]
And the constraints are:
\[ x \neq 0, , x \neq -1 \]