Question

Using the partial fractions technique, the function f(x)=(x³/x⁴)+37 x²+36, can be written as a sum of partial fractions

f(x)=x³/x⁴+37 x²+36=___

Answer 1

To write the function f(x) = (x³/x⁴)+37 x²+36 as a sum of partial fractions, factor the denominator and equate the coefficients of the like terms.

Step-by-step explanation:

To write the function f(x) = (x³/x⁴)+37 x²+36 as a sum of partial fractions, we first factor the denominator. The denominator x⁴+37x²+36 can be factored as (x²+36)(x²+1). The first fraction has a denominator of x²+1, and the second fraction has a denominator of x²+36.

Using the method of partial fractions, we write the function as: f(x) = A/(x²+1) + B/(x²+36), where A and B are constants.

To find A and B, we multiply both sides of the equation by the denominator and equate the coefficients of the like terms.

After solving the system of equations, we find that A = 0 and B = 1/6. Therefore, the function can be written as f(x) = 1/6/(x²+36).