Simplify the following rational expressions (don’t forget to cancel if you can). And define the constraints (what 𝓧 cannot be) (show all work):

Question

6x/x+1 - 5/5x^3

Answer 1

To simplify the expression \( \frac{6x}{x+1} - \frac{5}{5x^3} \), we will first find a common denominator and then combine the fractions.

Step 1: Identify the common denominator

The denominators are \( x + 1 \) and \( 5x^3 \). The least common denominator (LCD) for these two expressions is:

\[ \text{LCD} = 5x^3(x + 1) \]

Step 2: Rewrite each fraction with the common denominator

Now we need to rewrite each fraction with the common denominator.

1. For the first fraction \( \frac{6x}{x+1} \):

\[ \frac{6x}{x+1} = \frac{6x \cdot 5x^3}{(x+1) \cdot 5x^3} = \frac{30x^4}{5x^3(x+1)} \]

2. For the second fraction \( \frac{5}{5x^3} \):

\[ \frac{5}{5x^3} = \frac{5 \cdot (x+1)}{5x^3 \cdot (x+1)} = \frac{5(x+1)}{5x^3(x+1)} = \frac{5x + 5}{5x^3(x + 1)} \]

Step 3: Combine the rewritten fractions

Now we can combine the two fractions over the common denominator:

\[ \frac{30x^4}{5x^3(x + 1)} - \frac{5x + 5}{5x^3(x + 1)} = \frac{30x^4 - (5x + 5)}{5x^3(x + 1)} \]

Step 4: Simplify the numerator

Now we simplify the numerator \( 30x^4 - (5x + 5) \):

\[ 30x^4 - 5x - 5 \]

This cannot be factored easily, so we leave it as is.

Step 5: Write the final expression

Thus, the simplified expression is:

\[ \frac{30x^4 - 5x - 5}{5x^3(x + 1)} \]

Step 6: Define constraints for \( x \)

To define the restrictions on \( x \), we need to consider the denominators:

1. The term \( x + 1 \) cannot be zero:

   • \( x + 1 \neq 0 \Rightarrow x \neq -1 \)

2. The term \( 5x^3 \) cannot be zero:

   • \( 5x^3 \neq 0 \Rightarrow x \neq 0 \)

Constraints

The values that \( x \) cannot take are:

\[ x \neq 0 \quad \text{and} \quad x \neq -1 \]

Final Answer

The simplified expression is:

\[ \frac{30x^4 - 5x - 5}{5x^3(x + 1)} \quad \text{with constraints} \quad x \neq 0, x \neq -1 \]