Partial Fraction Decomposition Calculator

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Partial Fraction Decomposition Calculator
Distinct Linear Factors: (nx + m) / (ax² + bx + c)
Numerator: P(x) = n(x) + m
Denominator: Q(x) = ax² + bx + c
Result:
Enter coefficients to see decomposition.
function calculateDecomposition(){var nx=parseFloat(document.getElementById('nx').value);var nc=parseFloat(document.getElementById('nc').value);var da=parseFloat(document.getElementById('da').value);var db=parseFloat(document.getElementById('db').value);var dc=parseFloat(document.getElementById('dc').value);var stepsDiv=document.getElementById('showSteps').checked;var ans=document.getElementById('answer');if(isNaN(nx)||isNaN(nc)||isNaN(da)||isNaN(db)||isNaN(dc)){ans.innerHTML='Please enter valid numeric coefficients.';return;}if(da===0){ans.innerHTML='Denominator must be a quadratic (a ≠ 0).';return;}var disc=(db*db)-(4*da*dc);if(disc<=0){ans.innerHTML='Denominator must have distinct real roots. Discriminant (b²-4ac) must be > 0.';return;}var r1=(-db+Math.sqrt(disc))/(2*da);var r2=(-db-Math.sqrt(disc))/(2*da);var A=(nx*r1+nc)/(da*(r1-r2));var B=(nx*r2+nc)/(da*(r2-r1));var r1_sign=r1>=0?'- ':'+ ';var r2_sign=r2>=0?'- ':'+ ';var r1_val=Math.abs(r1.toFixed(4));var r2_val=Math.abs(r2.toFixed(4));var resultHTML='
' + A.toFixed(4) + ' / (x ' + r1_sign + r1_val + ') + ' + B.toFixed(4) + ' / (x ' + r2_sign + r2_val + ')
';if(stepsDiv){resultHTML+='
';resultHTML+='Step 1: Factor Denominator
'+da+'x² + '+db+'x + '+dc+' = '+da+'(x – '+r1.toFixed(2)+')(x – '+r2.toFixed(2)+')

';resultHTML+='Step 2: Setup Equation
('+nx+'x + '+nc+') / Q(x) = A / (x – '+r1.toFixed(2)+') + B / (x – '+r2.toFixed(2)+')

';resultHTML+='Step 3: Solve for A and B
Using Heaviside Cover-up:
A = ['+nx+'('+r1.toFixed(2)+') + '+nc+'] / ['+da+'('+r1.toFixed(2)+' – '+r2.toFixed(2)+')] = '+A.toFixed(4)+'
B = ['+nx+'('+r2.toFixed(2)+') + '+nc+'] / ['+da+'('+r2.toFixed(2)+' – '+r1.toFixed(2)+')] = '+B.toFixed(4);resultHTML+='
';}ans.innerHTML=resultHTML;}

Using the Partial Fraction Decomposition Calculator

The partial fraction decomposition calculator is a specialized algebraic tool designed to break down complex rational expressions into a sum of simpler fractions. This process is essential for calculus students, particularly when dealing with integration and Laplace transforms.

To use this calculator, follow these simple steps:

Numerator Coefficients
Enter the 'n' and 'm' values for the top part of your fraction (nx + m).
Denominator Coefficients
Enter the coefficients for the quadratic denominator (ax² + bx + c). The calculator will automatically factor this quadratic for you.
Show Solution Steps
Check this box to see the underlying algebra, including the roots of the denominator and the Heaviside method application.

How Partial Fraction Decomposition Works

Decomposition is the reverse process of finding a common denominator. When you have a fraction where the denominator is a polynomial of higher degree than the numerator, you can often "decompose" it into simpler parts. The most common case handled by our partial fraction decomposition calculator involves distinct linear factors.

P(x) / [(x – r₁)(x – r₂)] = A / (x – r₁) + B / (x – r₂)

The calculator utilizes the following mathematical process:

  • Factoring: The denominator $Q(x)$ is factored into its roots using the quadratic formula.
  • Constraint: For distinct real roots, the discriminant ($b^2 – 4ac$) must be positive.
  • Heaviside Cover-up: The constants $A$ and $B$ are calculated by evaluating the numerator at the roots of the denominator factors.

Practical Example

Example: Decompose the fraction (3x + 5) / (x² + x – 2).

Step-by-step solution:

  1. Identify Coefficients: n=3, m=5, a=1, b=1, c=-2.
  2. Factor Denominator: x² + x – 2 factors into (x + 2)(x – 1). The roots are r₁ = -2 and r₂ = 1.
  3. Set up Equation: (3x + 5) / [(x+2)(x-1)] = A / (x+2) + B / (x-1).
  4. Solve for A: Cover (x+2) and plug in x = -2: [3(-2)+5] / [-2-1] = -1 / -3 = 0.3333.
  5. Solve for B: Cover (x-1) and plug in x = 1: [3(1)+5] / [1-(-2)] = 8 / 3 = 2.6667.
  6. Final Result: 0.3333 / (x + 2) + 2.6667 / (x – 1).

Common Questions

What are the different types of partial fractions?

There are four main types: distinct linear factors (what this calculator focuses on), repeated linear factors, irreducible quadratic factors, and repeated irreducible quadratic factors. Each requires a slightly different algebraic setup.

When is decomposition not possible?

If the degree of the numerator is equal to or greater than the degree of the denominator, the fraction is "improper." You must first perform polynomial long division before using a partial fraction decomposition calculator on the remainder.

Why do we use this in calculus?

Integrating a complex rational function like 1/(x²-1) is difficult, but integrating the sum of 0.5/(x-1) and -0.5/(x+1) is simple, as they both result in natural logarithms (ln).

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