Question

What is the form of the partial fraction decomposition of a rational function?

1) Linear factors only
2) Quadratic factors only
3) Linear and quadratic factors
4) Cubic factors only

Answer 1

The partial fraction decomposition of a rational function usually involves both linear and quadratic factors. These factors help to break the function down into simpler components for easier manipulation in calculus.

Step-by-step explanation:

The partial fraction decomposition of a rational function typically includes linear and quadratic factors, which allows us to write the function as a sum of simpler fractions easier to integrate or differentiate. In cases where the denominator's factors are irreducible and of degree higher than two, quadratic terms will be used in the decomposition. The decomposition will break down the rational function into a sum where each term has a polynomial of a lower degree in the numerator than in the denominator.

For denominators with non-repeated linear factors, we have terms like A/(ax+b). If the denominator has repeated linear factors, we may have terms like A/(ax+b)^n. For non-repeated irreducible quadratic factors, terms will look like (Ax+B)/(ax^2+bx+c), and for repeated irreducible quadratic factors, we get (Ax+B)/(ax^2+bx+c)^n.