Question

Find the partial fraction decomposition of the rational function. {x+6} {x(x+3)}

Answer 1

The partial fraction decomposition of {x+6} divided by {x(x+3)} is found by expressing the function as A/x + B/(x+3) and solving for the constants A and B through a series of algebraic steps.

Step-by-step explanation:

The question is asking for the partial fraction decomposition of the rational function given by {x+6} divided by {x(x+3)}. To find the partial fractions, we assume that the function can be written as A/x + B/(x+3), where A and B are constants that we need to find. Multiplying both sides of the equation by the common denominator x(x+3) will give us a straightforward way to solve for A and B.

Here are the steps to find the constants A and B:

1. Multiply through by the common denominator x(x+3) to clear the fractions.
2. This yields x+6 = A(x+3) + Bx.
3. Expand the right side and collect like terms.
4. Equate the coefficients of like powers of x from both sides of the equation.
5. Solve the resulting system of equations for A and B.

Once A and B are found, the original function will be expressed as the sum of these simpler fractions, completing the partial fraction decomposition.