Question

Write the partial fraction decomposition for the expression. $$ \frac{3 x^{2}-x+1}{x(x+1)^{2}} $$

Short answer

The partial fraction decomposition of the expression \(\frac{3x^{2}-x+1}{x (x+1)^{2}}\) is \(\frac{A}{x} + \frac{Bx + C}{(x+1)^{2}}\), where values of A, B, and C are found by solving the polynomial equation created in step 4.

Step-by-step solution

Identify the denominator's factors

You must first factorize the denominator thoroughly. In this case, the expression \((x(x+1)^{2})\) has already been factorized and has two unique factors: \(x\) and \((x+1)^2\).

Set up the partial fractions

Partial fractions are set up based on the factors of the denominator. The aim is to write the original fraction as sum of these fractions. You must bear in mind that the numerator of each fraction is always one degree less than the denominator. In this case, the factors are \(x\) with degree 1 and \((x + 1)^{2}\) with degree 2. Hence, the partial fractions set up would be \(\frac{A}{x}\) and \(\frac{Bx + C}{(x+1)^{2}}\). So, \(\frac{3x^{2}-x+1}{x(x+1)^{2}} = \frac{A}{x} + \frac{Bx + C}{(x+1)^{2}}\).

Clear out the fractions

To get rid of the denominators in the equation, you multiply both sides of the equation by the original denominator: \(x(x+1)^{2}\). Doing this, you get: \(3x^{2}-x+1 = Ax(x+1)^{2} + Bx(x) + C(x)\).

Solve for A, B, and C

Collect terms to write the equation as a polynomial: \(Ax^{3} + 2Ax^{2} + Ax + Bx^2 + Cx = 3x^{2} - x + 1\). Here, to get the values of A,B,C you could equate coefficients of powers of x, or substitute suitable values of x to get system of linear equations and solve those equations. Choosing convenient values of x can solve the equation system.

Substitute values of A, B, C into partial fractions

Once you have the values of A, B, and C from Step 4, substitute those values back in to the equations for the partial fractions found in Step 2: \(\frac{3x^{2}-x+1}{x(x+1)^{2}}\) = \(\frac{A}{x} + \frac{Bx + C}{(x+1)^{2}}\) and simplify. This will result in the final answer.