Question

Write the partial fraction decomposition for the expression. $$ \frac{10 x+3}{x^{2}+x} $$

Short answer

The partial fraction decomposition of the given expression is \( \frac{3}{x} + \frac{7}{x + 1} \).

Step-by-step solution

Break down the expression into partial fractions

When dealing with a fraction whose denominator is a polynomial that can be factored, the first step is to write the fraction as a sum of simpler fractions. For the given exercise, the denominator \(x^{2} + x\) can be factored as \(x(x + 1)\). Therefore, we write the expression as follows: \[\frac{10x + 3}{x(x + 1)} = \frac{A}{x} + \frac{B}{x + 1}\] for some constants A and B.

Obtain common denominators and simplify

We multiply both sides of the equation by the common denominator \(x(x + 1)\) to get rid of the fractions. Doing this gives us: \[10x + 3 = A(x + 1) + Bx\]

Set up system of equations to find constants A and B

We can find the values of A and B by equating coefficients of like powers of x on both sides of the equation. This gives us: \(10 = B + A\) for the coefficient of x and \(3 = A\) for the constant term. Therefore, we have two equations, A = 3 and B + 3 = 10.

Solve for constants A and B

Solving for A and B gives us A = 3 and B = 10 - 3 = 7. Therefore, the decomposition of the original fraction is: \[\frac{10x + 3}{x(x + 1)} = \frac{3}{x} + \frac{7}{x + 1}\]