Partial Fraction Decomposition Calculator — Turn Rational Functions into Easy Pieces

A simple yet useful trick, partial fraction decomposition can simplify a complex rational expression by converting it into a sum of simple fractions. In that way, integration, inverse Laplace transforms, and simplification of algebra are much easier.

The calculator operates as explained below; a comprehensive worked example with your input is shown, there are special cases (repeated factors, improper fractions, irreducible quadratics) and hints on understanding the Fraction and Decimal output forms.

What the calculator expects (input format)

• Numerator (N(x)) — a polynomial, e.g. x^2 + 3x + 2.

• Denominator (D(x)) — another polynomial, e.g. x^3 - x^2 - 4x + 4.

• Output format — choose Fraction (exact symbolic decomposition) or Decimal (numeric values for coefficients).

Make sure to enter polynomials in standard algebraic form. The calculator will:

1. Numerator Check degree numerator less than degree denominator.

2. Otherwise, do a long division of the polynomials.

3. Factor and then attempt to set up the appropriate partial-fraction template and solve equations to get coefficients, provided that these factors are over the reals (when possible).

4. Output fraction form (symbolic), or rounded off decimal form (coefficients).

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Quick refresher: When and why use partial fractions

Partial fraction decomposition rewrites

N(x)D(x)\frac{N(x)}{D(x)}D(x)N(x)​

as a combination of other simpler terms such as Ax−a,distribution of Ax/x-a,x−aA, B(x−a)2,distribution of B/x-a, B(x-a)2, B or Cx+Dx2,bx+c,distribution of Cx+D. This is handy as every simple term can be readily incorporated/inverted in transforms.

Use it when you:

• Need to integrate rational functions ∫N(x)D(x) dx\int \frac{N(x)}{D(x)}\,dx∫D(x)N(x)​dx.

• Want to perform inverse Laplace transforms.

• Need a simplified symbolic form for algebra or partial simplification.

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Worked example (your exact input)

Input

• Numerator N(x)=x2+3x+2N(x) = x^2 + 3x + 2N(x)=x2+3x+2

• Denominator D(x)=x3−x2−4x+4D(x) = x^3 – x^2 – 4x + 4D(x)=x3−x2−4x+4

• Output format: Fraction (and we’ll also show Decimal)

Step 1 — Check properness

Degree of numerator = 2, degree of denominator = 3 → already a proper rational function. No polynomial division required.

Step 2 — Factor the denominator

Factor x3−x2−4x+4x^3 – x^2 – 4x + 4x3−x2−4x+4.

Group terms:

x2(x−1)−4(x−1)=(x−1)(x2−4)=(x−1)(x−2)(x+2).x^2(x-1)-4(x-1)=(x-1)(x^2-4)=(x-1)(x-2)(x+2).x2(x−1)−4(x−1)=(x−1)(x2−4)=(x−1)(x−2)(x+2).

So the denominator factors into three distinct linear factors: $(x-1)(x-2)(x+2)$.

Step 3 — Set up partial fraction form

Because these are distinct linear factors, the decomposition has this form:

x2+3x+2(x−1)(x−2)(x+2)=Ax−1+Bx−2+Cx+2.\frac{x^2+3x+2}{(x-1)(x-2)(x+2)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x+2}.(x−1)(x−2)(x+2)x2+3x+2​=x−1A​+x−2B​+x+2C​.

Step 4 — Solve for coefficients

Divide both sides by the denominator: x2+3x+2 = A(x12-2)(x+2) + B(x12-1)(x+2) + C(x12-1)(x12-2).x2+3x+ 2= A(x-2)X +2 + B(x-1)X +2 + C(x-1)X +2.

Plug in convenient $x$-values (the roots) to solve quickly:

• $x=1:1+3+2=6=A(1-2)(1+2)=A(-1)(3)=-3A\Rightarrow A=-2.$

• $x=2:4+6+2=12=B(2-1)(2+2)=4B\Rightarrow B=3.$

• $x=-2:4-6+2=0=C(-2-1)(-2-2)=12C\Rightarrow C=0.$

So the decomposition is

x2+3x+2×3−x2−4x+4=−2x−1+3x−2+0x+2\boxed{\frac{x^2+3x+2}{x^3-x^2-4x+4}=-\frac{2}{x-1}+\frac{3}{x-2}+\frac{0}{x+2}}x3−x2−4x+4x2+3x+2​=−x−12​+x−23​+x+20​​

The term with $C$ vanishes, so in simplified form:

−2x−1+3x−2.-\frac{2}{x-1}+\frac{3}{x-2}.−x−12​+x−23​.

Fraction output (exact):

−2x−1+3x−2.-\dfrac{2}{x-1}+\dfrac{3}{x-2}.−x−12​+x−23​.

Decimal output (coefficients as decimals):
$A=-2.0,B=3.0,C=0.0$, so the same expression numerically:

−2.0x−1+3.0x−2.-\frac{2.0}{x-1} + \frac{3.0}{x-2}.−x−12.0​+x−23.0​.

Integration example (bonus)
One common use is integration:

∫x2+3x+2×3−x2−4x+4 dx=∫(−2x−1+3x−2) dx=−2ln⁡∣x−1∣+3ln⁡∣x−2∣+C.\int \frac{x^2+3x+2}{x^3-x^2-4x+4}\,dx = \int\left(-\frac{2}{x-1}+\frac{3}{x-2}\right)\,dx = -2\ln|x-1| + 3\ln|x-2| + C.∫x3−x2−4x+4x2+3x+2​dx=∫(−x−12​+x−23​)dx=−2ln∣x−1∣+3ln∣x−2∣+C.

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Special cases the calculator handles

1. Improper fractions (degree numerator ≥ degree denominator)

   • The calculator first performs polynomial long division:

     N(x)D(x)=Q(x)+R(x)D(x),\frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)},D(x)N(x)​=Q(x)+D(x)R(x)​,

     then decomposes the proper remainder R(x)D(x)\frac{R(x)}{D(x)}D(x)R(x)​.

2. Repeated linear factors (e.g. (x−a)2(x-a)^2(x−a)2)

   • Template becomes:

     Ax−a+B(x−a)2+⋯\frac{A}{x-a} + \frac{B}{(x-a)^2} + \cdotsx−aA​+(x−a)2B​+⋯

     with terms up to the repeated power.

3. Irreducible quadratic factors (e.g. x2+1x^2+1x2+1 that doesn’t factor over reals)

   • Template uses linear numerators:

     Cx+Dx2+1.\frac{Cx+D}{x^2+1}.x2+1Cx+D​.

4. Complex roots / nonreal factorization

   • For real-valued outputs the calculator keeps irreducible quadratic blocks with linear numerators (real partial fractions). If asked, a complex decomposition can be done but most users want real partial fractions.

5. Symbolic vs numeric

   • Fraction mode yields coefficient ratios that are precise fractions (e.g. −2-2−2, 333, 12\tfrac1221).

   • Decimal mode gives floating-points approximations (e.g. -2.0000-2.0000-2.0000, 3.00003.00003.0000) as rounded off.

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Tips for entering polynomials

• Use x as the variable, no spaces required but they’re fine: x^3-x^2-4x+4.

• For coefficients: -2x^2 or -2*x^2 both work (calculator accepts standard algebraic notation).

• If the numerator is improper, don’t worry — you can still paste it; the tool will divide automatically.

• If the denominator doesn’t factor nicely over reals, choose Fraction mode to get symbolic linear-over-quadratic pieces, or Decimal to see numeric coefficients.

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Applications in calculus, ODEs, and engineering

• Integration. Once decomposed, each term integrates to a simple logarithm or arctan form.

• Inverse Laplace transforms. The transform of rational functions is handled using partial fractions to match transform tables.

• Control theory / signal processing. Transfer functions often decompose into simple poles for stability analysis.

• Algebraic simplification. Rewriting rational functions as a sum of simpler terms clarifies behavior near poles.

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Common pitfalls and how the calculator avoids them

• Forgetting to divide when numerator degree ≥ denominator degree. The calculator does this automatically.

• Assuming distinct linear factors when there are repeats. The correct form is inferred from factor multiplicity.

• Rounding too aggressively in Decimal mode. Use Fraction mode for exactness; use higher precision if you need numeric approximations.

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Frequently asked questions (short)

Q: What if my denominator won’t factor nicely?
A: The tool leaves irreducible quadratics as linear numerators over those quadratics (real decomposition). For complex roots you can request complex coefficients.

Q: Can the calculator show step-by-step work?
A: Some implementations show factoring, substitution, and solving steps. If you need step-by-step, choose the verbose or steps option (if available).

Q: How accurate is Decimal output?
A: Decimal output is as accurate as the chosen rounding precision—use Fraction mode for exact symbolic results.

Q: Will the calculator integrate the result for me?
A: Many calculators offer integration of the decomposed terms. If present, you’ll get $\int$ results like $-2\ln |x-1|+3\ln |x-2|+C$.

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Closing thoughts

Partial fraction decomposition transforms a single complicated rational expression into a handful of tidy terms you can integrate, invert, or analyze easily. Your provided example,

x2+3x+2×3−x2−4x+4,\frac{x^2+3x+2}{x^3-x^2-4x+4},x3−x2−4x+4x2+3x+2​,

decomposes cleanly to

−2x−1+3x−2,-\frac{2}{x-1}+\frac{3}{x-2},−x−12​+x−23​,

which is about as elegant and useful as partial fractions get.

If you want, I can also:

• Produce a step-by-step printable PDF of the decomposition for that example, or

• Provide code snippets (Python / SymPy) that reproduce the calculator’s logic, or

• Generate a short video script showing how to enter the polynomials and interpret Fraction vs Decimal output.

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