Final answer:
The first step in dealing with partial fractions is to factorize the denominators. This initial step is crucial as it allows you to identify the form of the partial fraction expansion and solve for unknown coefficients in the next steps. Option A is the correct answer.
Step-by-step explanation:
When dealing with partial fractions, the first step you should always take is to factorize the denominators of the expression you are working with. This is essential because the method of partial fractions involves breaking a complex fraction into a sum of simpler fractions, where the denominators are typically polynomial factors of the original denominator. These factors can be linear or irreducible quadratic polynomials. Factorizing the denominator makes it possible to identify the correct form of the partial fraction expansion, which you will then solve for.
Once the denominator is factorized, you can then proceed with other steps, such as setting up a system of equations to solve for the unknown coefficients in the partial fractions. If you encounter complex fractions with common denominators, you may temporarily skip factorization, but only for the sake of simplifying before factorizing. This strategy ensures that any simplifications won't overlook possible factorization routes, which are vital to the correct application of the method of partial fractions.
Only after completely factorizing the original fraction's denominator and rewriting the fraction as a sum of partial fractions with unresolved coefficients do you multiply every term by the product of the denominators to clear the fractions and solve for the unknowns. Always remember, factorization is the fundamental first step without which the process cannot proceed.
In conclusion, the correct answer to the question is a) Factorize the denominators.