Rational to Partial Fraction Converter
Rational to Partial Fraction Converter decomposes a rational function \\( \frac{P(x)}{Q(x)} \\) into its partial fraction form, displaying steps with MathJax.
Rational to Partial Fraction Converter
Rational to Partial Fraction Converter decomposes a rational function \\( \frac{P(x)}{Q(x)} \\) into its partial fraction form. Input the numerator and denominator polynomials to see the decomposition with MathJax-rendered steps. Results are copyable, with sharing and embedding options for calculus and algebra students.
Example 1: Partial Fraction Decomposition with Distinct Linear Factors
Rational Function: \\( \frac{x + 1}{x^2 – 1} \\).
Step 1: Factor the denominator.
\\( x^2 – 1 = (x – 1)(x + 1) \\).
Step 2: Set up partial fractions.
\\( \frac{x + 1}{(x – 1)(x + 1)} = \frac{A}{x – 1} + \frac{B}{x + 1} \\).
Step 3: Solve for coefficients.
Multiply through: \\( x + 1 = A(x + 1) + B(x – 1) \\).
Simplify: \\( x + 1 = (A + B)x + (A – B) \\).
Equate coefficients: \\( A + B = 1 \\), \\( A – B = 1 \\).
Solve: \\( A = 1 \\), \\( B = 0 \\).
Step 4: Write the decomposition.
\\( \frac{x + 1}{x^2 – 1} = \frac{1}{x – 1} \\).
Step 5: Conclusion.
The partial fraction decomposition is \\( \frac{1}{x – 1} \\).
Example 2: Partial Fraction Decomposition with Repeated Linear Factors
Rational Function: \\( \frac{2x + 1}{x^2 (x – 1)} \\).
Step 1: Factor the denominator.
\\( Q(x) = x^2 (x – 1) \\).
Step 2: Set up partial fractions.
\\( \frac{2x + 1}{x^2 (x – 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x – 1} \\).
Step 3: Solve for coefficients.
Multiply through: \\( 2x + 1 = A x (x – 1) + B (x – 1) + C x^2 \\).
Simplify: \\( 2x + 1 = (A + C)x^2 + (-A + B)x – B \\).
Equate coefficients: \\( A + C = 0 \\), \\( -A + B = 2 \\), \\( -B = 1 \\).
Solve: \\( B = -1 \\), \\( A = -3 \\), \\( C = 3 \\).
Step 4: Write the decomposition.
\\( \frac{2x + 1}{x^2 (x – 1)} = -\frac{3}{x} – \frac{1}{x^2} + \frac{3}{x – 1} \\).
Step 5: Conclusion.
The partial fraction decomposition is \\( -\frac{3}{x} – \frac{1}{x^2} + \frac{3}{x – 1} \\).