유리식 계산기
이 계산기는 (x² − 9)/(x² + 5x + 6) 같은 유리식을 약분하고, 두 유리식을 더하거나 빼거나 곱하거나 나눕니다. 모든 분자와 분모를 인수분해하고, 공통인수를 약분하며, 제외해야 하는 x의 값을 알려 줍니다.
결과
값을 입력하고 계산을 누르면 결과가 표시됩니다.
공식
Adding or subtracting: use a common denominator (the product bd works; the calculator cancels any common factor afterwards).
Multiplying and dividing: multiply straight across; to divide, multiply by the reciprocal.
이 계산기 사용법
- Choose an operation: simplify one rational expression, or add, subtract, multiply or divide two.
- Type each numerator and denominator as a polynomial in one variable, e.g. x^2 - 9 or (x + 1)(x - 2).
- Press Calculate. You get the simplified expression in factored form, the excluded values, and every step.
풀이 예제
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).
자주 하는 실수
- 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.
자주 묻는 질문
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.