(a) For the function x^3/(x^2 + 6x + 5), the denominator can be factored into (x+1)(x+5). The form of the partial fraction decomposition would be:

A/(x+1) + B/(x+5)

(b) For the function (6x + 1)/[(x + 1)^3 * (x^2 + 4)^2], the form of the partial fraction decomposition would be:

A/(x+1) + B/(x+1)^2 + C/(x+1)^3 + D/(x^2+4) + E/(x^2+4)^2

In both cases, A, B, C, D, and E are constants that can be determined by equating coefficients and solving the resulting system of equations.

Question

(a) For the function x^3/(x^2 + 6x + 5), the denominator can be factored into (x+1)(x+5). The form of the partial fraction decomposition would be:

A/(x+1) + B/(x+5)

(b) For the function (6x + 1)/[(x + 1)^3 * (x^2 + 4)^2], the form of the partial fraction decomposition would be:

A/(x+1) + B/(x+1)^2 + C/(x+1)^3 + D/(x^2+4) + E/(x^2+4)^2

In both cases, A, B, C, D, and E are constants that can be determined by equating coefficients and solving the resulting system of equations.

Knowee AI · Accepted Answer

(a) For the function x^3/(x^2 + 6x + 5), the denominator can be factored into (x+1)(x+5). The form of the partial fraction decomposition would be:

A/(x+1) + B/(x+5)

(b) For the function (6x + 1)/[(x + 1)^3 * (x^2 + 4)^2], the form of the partial fraction decomposition would be:

A/(x+1) + B/(x+1)^2 + C/(x+1)^3 + D/(x^2+4) + E/(x^2+4)^2

In both cases, A, B, C, D, and E are constants that can be determined by equating coefficients and solving the resulting system of equations.

Write out the form of the partial fraction decomposition of the function (see example). Do not determine the numerical values of the coefficients.(a) x3x2 + 6x + 5 (b) 6x + 1(x + 1)3(x2 + 4)2