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Калькулятор рациональных выражений

Этот калькулятор упрощает алгебраические дроби вроде (x² − 9)/(x² + 5x + 6), а также складывает, вычитает, умножает или делит две такие дроби. Он раскладывает каждый числитель и знаменатель, сокращает общие множители и выписывает значения x, которые нужно исключить.

Результат

Введите значения и нажмите «Вычислить», чтобы увидеть результат.

Формула

ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Adding or subtracting: use a common denominator (the product bd works; the calculator cancels any common factor afterwards).

ab×cd=acbd,ab÷cd=ab×dc\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}, \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Multiplying and dividing: multiply straight across; to divide, multiply by the reciprocal.

Как пользоваться этим калькулятором

  1. Choose an operation: simplify one rational expression, or add, subtract, multiply or divide two.
  2. Type each numerator and denominator as a polynomial in one variable, e.g. x^2 - 9 or (x + 1)(x - 2).
  3. Press Calculate. You get the simplified expression in factored form, the excluded values, and every step.

Разобранные примеры

Easy: simplifying a rational expression

Factor the top and bottom: x² − 9 = (x − 3)(x + 3) and x² + 5x + 6 = (x + 2)(x + 3).

Cancel the common factor (x + 3): the expression simplifies to (x − 3)/(x + 2).

Excluded values come from the original denominator: x ≠ −3 and x ≠ −2.

Medium: adding with different denominators

Use the common denominator (x + 1)(x − 1).

Numerator: 1·(x − 1) + 2·(x + 1) = 3x + 1.

So 1/(x + 1) + 2/(x − 1) = (3x + 1)/((x + 1)(x − 1)), with x ≠ −1 and x ≠ 1.

Hard: dividing by a second fraction

Multiply by the reciprocal: (x² − 4)/(x + 3) × (x² + 3x)/(x − 2).

Factor: (x − 2)(x + 2)/(x + 3) × x(x + 3)/(x − 2). Cancel (x − 2) and (x + 3).

The result is x(x + 2). Excluded: x ≠ −3 and x ≠ 0 (denominators) and x ≠ 2 (the divisor would be 0).

Частые ошибки

  • Cancelling terms instead of factors: in (x + 3)/(x + 5) nothing cancels — only whole factors can be cancelled.
  • Adding numerators and denominators straight across: 1/x + 1/y is (x + y)/(xy), not 2/(x + y).
  • Forgetting excluded values that disappear when you cancel: (x² − 9)/(x + 3) is x − 3 only for x ≠ −3.
  • Dividing without flipping the second fraction: ÷ c/d means × d/c.

Частые вопросы

How do you simplify a rational expression?

Factor the numerator and the denominator completely, then cancel any factor that appears in both. Note the values of x that make the original denominator zero — they stay excluded.

How do I add or subtract rational expressions?

Write both over a common denominator, add or subtract the numerators, then simplify. For 1/(x + 1) + 2/(x − 1) the common denominator is (x + 1)(x − 1) and the result is (3x + 1)/((x + 1)(x − 1)).

How do I divide rational expressions?

Multiply the first expression by the reciprocal of the second, then factor and cancel. The values that make the second numerator zero are excluded too.

What are excluded values?

Values of the variable that make a denominator zero, so the expression is undefined there. They are listed even when the factor that causes them cancels.

What can I type?

Polynomials in one variable: numbers, x, + − × and powers, with brackets — for example 2x^2 - 8 or (x + 1)^2. Degrees up to 6.

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