Step by step fraction decomposition sample

Step by step solution of fraction decomposition, received by our online calculator is presented below:

Partial fraction decomposition calculator:

Find partial fraction decomposition:

\(\dfrac{x^2+4\cdot x-1}{x \cdot (x-1) \cdot (x^2+x-1)}\)

Input interpretation

Find partial fraction decomposition:

\(\dfrac{x^2+4\cdot x-1}{x \cdot (x-1) \cdot (x^2+x-1)}\)

Answer:

\(-\dfrac{1}{x}+\dfrac{4}{x-1}-\dfrac{3\cdot x+6}{x^2+x-1}\)

Step by step solution:

Step 1

Since the degree of the numeration is less than the degree of the denominator, the original fraction is a proper one. We express its partial fraction decomposition as a sum of fractions in the general form (with undetermined coefficients for now).

\(\dfrac{x^2+4\cdot x-1}{x \cdot (x-1) \cdot (x^2+x-1)}=\dfrac{A}{x}+\dfrac{B}{x-1}+\dfrac{C\cdot x+D}{x^2+x-1}\)

Step 2

To determine the unknown coefficients in the decomposition, we multiply both sides of the expression by the common denominator.

\(x^2+4\cdot x-1=A\cdot\left(x-1\right)\cdot\left(x^2+x-1\right)+B\cdot x\cdot \left(x^2+x-1\right)+\left(C\cdot x+D\right)\cdot x\cdot\left(x-1\right)\)

Step 3

Determine the coefficients of the corresponding powes of \(x\):

\(x^2+4\cdot x-1=A+\left(-2\cdot A-B-D\right)\cdot x+\left(B-C+D\right)\cdot x^2+\left(A+B+C\right)\cdot x^3\)

Step 4

To maintain true equality, we should equate the coefficients for same powers of \(x\):

\(\begin{cases}-1=A \\ 4=-2\cdot A-B-D \\ 1=B-C+D \\ 0=A+B+C\end{cases}\)

Step 5

By solving the system of linear equations, we obtain the values of the unknown coefficients:

\(\begin{cases}A=-1 \\ B=4 \\ C=-3 \\ D=-6\end{cases}\)

Step 6

Substituting the values of the unknown coefficients into the decomposition, we get the final form of the expansion:

\(\dfrac{x^2+4\cdot x-1}{x \cdot (x-1) \cdot (x^2+x-1)}=-\dfrac{1}{x}+\dfrac{4}{x-1}-\dfrac{3\cdot x+6}{x^2+x-1}\)

Install calculator on your site

As you can see, our online fraction decomposition calculator returns step by step solution. The solution is clear and easily can be understand by any student, so don't lose you time and use our online calculator to get step by step fraction decomposition solution.

Step by step fraction decomposition sample

Step by step solution:

See also:

Leave your comment: