Question

implify the following rational expressions (don’t forget to cancel if you can). And define the constraints (what 𝓧 cannot be) (show all work): 6x/x+1 - 5/5x^3

Answer 1

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:

1. \(x^3 = 0 \implies x = 0\)
2. \(x + 1 = 0 \implies x = -1\)

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:

\[
x \neq 0, \, x \neq -1
\]