多項式の計算機：零点・因数・標準形
この多項式ソルバーは多項式を展開して標準形に整理し、因数分解して零点を正確に求めます。整数や分数、√2 のような無理数、i のような複素数の零点にも対応します。(x + 2)(x − 3)^2 や x^3 − 6x^2 + 11x − 6 のように入力してください。
結果
値を入力して「計算」を押すと結果が表示されます。
公式
Standard form: like terms combined, powers in descending order.
Rational root theorem: any rational zero p/q has p dividing the constant term and q dividing the leading coefficient.
この計算ツールの使い方
- Type a polynomial in one variable. Brackets and powers are expanded for you, so (x + 1)^3 or 2(x - 1)(x + 4) both work.
- Press Calculate to get the standard form, the factored form, every zero and the degree.
- Zeros that are not whole numbers are shown exactly (fractions, √ or i) with decimals underneath.
解き方付き例題
Expand a polynomial and find its zeros
Expand: (x − 3)² = x² − 6x + 9, then (x + 2)(x² − 6x + 9) = x³ − 4x² − 3x + 18.
The factored form shows the zeros: x = −2, and x = 3 twice (multiplicity 2).
Degree 3, so there are three zeros counted with multiplicity.
Zeros of a cubic by the rational root theorem
Candidates are ±1, ±2, ±3, ±6. x = 1 gives 1 − 6 + 11 − 6 = 0.
Divide by (x − 1): x² − 5x + 6 = (x − 2)(x − 3).
Zeros: 1, 2 and 3.
Irrational zeros
Group: x²(x − 2) − 2(x − 2) = (x − 2)(x² − 2).
x² − 2 = 0 gives x = ±√2 ≈ ±1.414214.
Zeros in order: −√2, √2, 2.
Complex zeros
Difference of squares twice: x⁴ − 1 = (x² − 1)(x² + 1) = (x + 1)(x − 1)(x² + 1).
x² + 1 = 0 has no real solution: x = ±i.
Zeros: −1, 1, i and −i.
よくある間違い
- Squaring a bracket term by term: (x − 3)² is x² − 6x + 9, not x² + 9.
- Dropping a sign when expanding: −2(x − 4) is −2x + 8.
- Forgetting repeated zeros: in (x − 3)², x = 3 counts twice.
- Stopping at real zeros: a degree-4 polynomial always has four zeros counted with multiplicity, some may be complex.
よくある質問
How do you find the zeros of a polynomial?
Factor it, then set each factor to zero. For a cubic or higher, test the candidates from the rational root theorem, divide out each zero you find, and finish the last quadratic with the quadratic formula.
What is standard form of a polynomial?
All brackets expanded, like terms combined, and the terms written from the highest power down — for example x³ − 4x² − 3x + 18.
Can it add, subtract and simplify polynomials?
Yes — type the whole expression, such as (2x^2 + 3x) - (x^2 - 5x + 1), and the standard form is the simplified result: x² + 8x − 1.
Why are some zeros missing?
If a polynomial has a factor of degree 3 or more with no rational zeros, its zeros can’t always be written exactly; the calculator names that factor so you can graph it or use a numerical method.
What is the largest degree allowed?
Degree 8 after expanding, with coefficients up to about a billion.