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Problem 1

# As a single rational expression, simplified as much as possible. $$\frac{x-4}{x+1} \cdot \frac{2 x+1}{x-1}$$

Expert verified
As a single rational expression, simplified as much as possible, we have: $$\frac{x - 4}{x + 1} \cdot \frac{2x + 1}{x - 1} = \frac{2x^2 - 7x - 4}{x^2 - 1}$$.
See the step by step solution

## Step 1: Multiplying the numerators

To multiply the rational functions, we need to multiply the numerators together and the denominators together. In this case, we have $$(x - 4) \cdot (2x + 1)$$ for the numerator of the result.

## Step 2: Expanding the numerator

Now we will expand the multiplication in the numerator by applying the distributive property: $(x - 4)(2x + 1) = x(2x + 1) - 4(2x + 1)$

## Step 3: Applying distributive property on the numerator

Now we apply the distributive property again on each part of the multiplication: $x(2x + 1) - 4(2x + 1) = 2x^2 + x - 8x - 4$

## Step 4: Simplifying the numerator

Combine the like terms in the numerator: $2x^2 + x - 8x - 4 = 2x^2 - 7x - 4$

## Step 5: Multiplying the denominators

Now for the denominator of the result, we simply multiply the two given denominators together: $$(x + 1)(x - 1)$$.

## Step 6: Expanding the denominator

We can use the difference of squares formula to expand the denominator: $(x + 1)(x - 1) = x^2 - 1$

## Step 7: Putting it all together

Combine the simplified numerator and denominator of the result to obtain the final expression: $\frac{2x^2 - 7x - 4}{x^2 - 1}$ So as a single rational expression, simplified as much as possible, we have: $\frac{x - 4}{x + 1} \cdot \frac{2x + 1}{x - 1} = \frac{2x^2 - 7x - 4}{x^2 - 1}$

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