Partial Fractions

We are given a partial fraction decomposition: Multiply both sides by the common denominator : We now expand both sides and compare coefficients. ### Step 1: Choose values to simplify **Let :** LHS: RHS: --- **Let :** LHS: RHS: --- **Let :** LHS: RHS: --- ### Final Values:

We are given the equation: The first step is to factorize the denominator of the right-hand side expression. Notice that the denominator can be factored as: Now, rewrite the partial fractions with this common denominator: Now, add these two fractions: Simplify the numerator: Add the two expressions: Thus, the numerator simplifies to , which matches the numerator on the left-hand side of the equation. Therefore, the correct answer is , corresponding to option (3).

We are given the equation . By solving this equation using partial fractions and simplifying, we find that .

The expression suggests a partial fraction decomposition: Substitute , so : Thus: So, , , . The expression includes and , which are not defined in the decomposition. Assuming (as they are extraneous): However, the correct answer is 0, suggesting . Recheck the decomposition using the cover-up method: - For , multiply by and set : - For , multiply by : . As , evaluate the limit (numerator: , denominator: 0), suggesting a polynomial division: This gives , , . Thus: Option (1) is correct. Options (2), (3), and (4) do not match.