Plany

Kalkulator ułamków prostych

Ten kalkulator rozkłada funkcję wymierną, taką jak (x + 5)/(x² + x − 2), na ułamki proste: 2/(x − 1) − 1/(x + 2). Obsługuje różne i powtarzające się czynniki liniowe oraz nierozkładalne czynniki kwadratowe, najpierw dzieli wielomiany, gdy ułamek jest niewłaściwy, i sprawdza odpowiedź, składając ją z powrotem.

Wynik

Wpisz wartości i naciśnij Oblicz, aby zobaczyć wynik.

Wzór

P(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}

Distinct linear factors: one constant over each factor.

P(x)(x−a)2=Ax−a+B(x−a)2\frac{P(x)}{(x - a)^2} = \frac{A}{x - a} + \frac{B}{(x - a)^2}

A repeated factor gets one term for each power.

P(x)(x−a)(x2+px+q)=Ax−a+Bx+Cx2+px+q\frac{P(x)}{(x - a)(x^2 + px + q)} = \frac{A}{x - a} + \frac{Bx + C}{x^2 + px + q}

An irreducible quadratic factor gets a linear numerator Bx + C.

Jak korzystać z tego kalkulatora

  1. Type the numerator and the denominator as polynomials, e.g. x + 5 and x^2 + x - 2.
  2. Press Calculate. The calculator divides first if the numerator’s degree is not smaller, factors the denominator, and solves for the unknown numerators.
  3. Read the decomposition and the steps; the check multiplies everything back together.

Rozwiązane przykłady

Easy: two distinct linear factors

Factor the denominator: x² + x − 2 = (x + 2)(x − 1).

Write (x + 5)/((x + 2)(x − 1)) = A/(x + 2) + B/(x − 1), so x + 5 = A(x − 1) + B(x + 2).

x = 1 gives 6 = 3B, so B = 2; x = −2 gives 3 = −3A, so A = −1.

Answer: −1/(x + 2) + 2/(x − 1).

Medium: a repeated factor

The denominator is (x + 1)², so use A/(x + 1) + B/(x + 1)².

3x + 5 = A(x + 1) + B. Comparing x terms: A = 3; constants: 3 + B = 5, so B = 2.

Answer: 3/(x + 1) + 2/(x + 1)².

Hard: a quadratic factor

Factor: x³ + x = x(x² + 1), and x² + 1 has no real roots, so it gets a numerator Bx + C.

2x² + 3 = A(x² + 1) + (Bx + C)x. x = 0 gives A = 3.

Then 2x² + 3 = (3 + B)x² + Cx + 3, so B = −1 and C = 0.

Answer: 3/x − x/(x² + 1).

Improper fraction: divide first

The numerator’s degree is not smaller, so divide: x³ = x(x² − 1) + x, giving x + x/(x² − 1).

x/((x + 1)(x − 1)) = A/(x + 1) + B/(x − 1): x = 1 gives B = 1/2 and x = −1 gives A = 1/2.

Answer: x + 1/(2(x + 1)) + 1/(2(x − 1)).

Częste błędy

  • Skipping long division: if the numerator’s degree is not smaller than the denominator’s, divide first.
  • Using only one term for a repeated factor: (x + 1)² needs both A/(x + 1) and B/(x + 1)².
  • Giving a quadratic factor a constant numerator: it needs Bx + C.
  • Not factoring the denominator completely before setting up the terms.

Najczęstsze pytania

What is partial fraction decomposition?

Rewriting one rational function as a sum of simpler fractions whose denominators are the factors of the original denominator. It is used to integrate rational functions and in Laplace transforms.

How do I find the constants A and B?

Multiply both sides by the denominator, then either substitute the roots of each factor (the cover-up method) or compare coefficients. The calculator solves the coefficient equations exactly.

Which denominators can this calculator handle?

Any polynomial up to degree 6 that factors into linear factors with rational roots (repeated or not) and at most one irreducible quadratic factor. It says so when a denominator falls outside that.

How is the answer checked?

All the fractions are multiplied back over the common denominator, and the new numerator must equal yours exactly.

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