طرح‌ها

ماشین‌حساب چندجمله‌ای: صفرها، عامل‌ها و شکل استاندارد

این ماشین‌حساب یک چندجمله‌ای را باز و به شکل استاندارد ساده می‌کند، آن را تجزیه می‌کند و صفرهایش را دقیق پیدا می‌کند — عددهای صحیح و کسرها، رادیکال‌هایی مثل √2 و صفرهای مختلطی مثل i. چیزی مثل (x + 2)(x − 3)^2 یا x^3 − 6x^2 + 11x − 6 بنویسید.

نتیجه

مقادیر را وارد کنید و برای دیدن نتیجه «محاسبه» را بزنید.

فرمول

anxn+an−1xn−1+⋯+a1x+a0a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0

Standard form: like terms combined, powers in descending order.

x=pq,p∣a0,  q∣anx = \frac{p}{q}, \quad p \mid a_0,\; q \mid a_n

Rational root theorem: any rational zero p/q has p dividing the constant term and q dividing the leading coefficient.

نحوه استفاده از این ماشین‌حساب

  1. 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.
  2. Press Calculate to get the standard form, the factored form, every zero and the degree.
  3. Zeros that are not whole numbers are shown exactly (fractions, √ or i) with decimals underneath.

مثال‌های حل‌شده

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.

اشتباه‌های رایج

  • 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.

پرسش‌های متداول

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.

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