Mathematics
Mathematics
Problem
Consider the rational function (i) Express it as the sum of a polynomial and a proper fraction. (ii) Write the general partial-fraction decomposition of the proper fraction obtained in part (i) (do not evaluate the constants).Intuitive explanation
The key here is polynomial long division followed by partial fractions. When a rational function's numerator has degree ≥ the denominator's degree, you can't jump straight to partial fractions—first divide to extract the polynomial part, leaving a proper fraction (numerator degree < denominator degree) that's ready for decomposition. Once you have that proper fraction, factor the denominator and set up the partial-fraction form by assigning unknown constants to each factor, which you'd solve for if needed.
Intuitive explanation
The key here is polynomial long division followed by partial fractions. When a rational function's numerator has degree ≥ the denominator's degree, you can't jump straight to partial fractions—first divide to extract the polynomial part, leaving a proper fraction (numerator degree < denominator degree) that's ready for decomposition. Once you have that proper fraction, factor the denominator and set up the partial-fraction form by assigning unknown constants to each factor, which you'd solve for if needed.