套餐

因式分解计算器

这个因式分解计算器可分解多项式和三项式：提取最大公因式，找出每个有理根对应的因式，并通过乘回去检验答案。

一个变量，次数不超过 6。用 ^ 表示乘方，例如 x^3 - 6x^2 + 11x - 6

结果

输入数值并点击“计算”查看结果。

公式

ax2+bx+c=ax2+mx+nx+cwhere mn=ac, m+n=bax^2 + bx + c = ax^2 + mx + nx + c \quad\text{where } mn = ac,\ m + n = b

The x method (ac method) for trinomials: split the middle term, then factor by grouping.

P ⁣(pq)=0  ⇒  (qx−p) is a factorP\!\left(\tfrac{p}{q}\right) = 0 \;\Rightarrow\; (qx - p) \text{ is a factor}

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

如何使用此计算器

  1. Type the polynomial with ^ for powers, for example 2x^2 + 7x - 4 or x^3 - 6x^2 + 11x - 6.
  2. Press Calculate. The factored form comes first, then the steps: common factor, the x method or possible roots, each factor found, and the check.
  3. Any factor that has no rational roots is left as it is and labelled.

例题

Easy: take out the common factor of 6x² + 9x

Both terms are divisible by 3 and by x, so the greatest common factor is 3x.

6x² ÷ 3x = 2x and 9x ÷ 3x = 3, so 6x² + 9x = 3x(2x + 3).

Check: 3x · 2x + 3x · 3 = 6x² + 9x ✓.

Medium: factoring a trinomial, 2x² + 7x − 4

x method: a·c = 2 × (−4) = −8. Two numbers that multiply to −8 and add to 7 are −1 and 8.

Split the middle term: 2x² − x + 8x − 4 = x(2x − 1) + 4(2x − 1) = (x + 4)(2x − 1).

Check: (x + 4)(2x − 1) = 2x² − x + 8x − 4 = 2x² + 7x − 4 ✓. The roots are x = −4 and x = 1/2.

Hard: a cubic, x³ − 6x² + 11x − 6

Possible rational roots divide the constant term 6: ±1, ±2, ±3, ±6.

P(1) = 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor; dividing leaves x² − 5x + 6.

x² − 5x + 6 = (x − 2)(x − 3), so x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3).

Check: (x − 1)(x − 2) = x² − 3x + 2, and (x² − 3x + 2)(x − 3) = x³ − 6x² + 11x − 6 ✓.

常见错误

  • Not taking out the common factor first. 2x³ − 8x = 2x(x² − 4) = 2x(x + 2)(x − 2); start with 2x and the rest is easier.
  • Stopping too early: factoring completely means no factor can be broken down further — x² − 4 still splits into (x + 2)(x − 2).
  • Sign mistakes in the x method: for 2x² + 7x − 4 the pair is 8 and −1 (product −8, sum 7), not −8 and 1.
  • Forgetting to check. Multiplying the factors back out should give the original polynomial exactly.

常见问题

What does it mean to factor a polynomial completely?

Writing the polynomial as a product where no factor can be split further: first the greatest common factor, then every smaller factor. For example 2x³ − 8x = 2x(x + 2)(x − 2).

How does the x method work for factoring trinomials?

For ax² + bx + c, find two numbers that multiply to a·c and add to b. Use them to split bx into two terms, then factor by grouping. Sumzo shows the two numbers whenever they exist.

Can the factor calculator handle polynomials of degree 3 or 4?

Yes, up to degree 6. It finds every factor with a rational root using the rational root theorem. A remaining factor of degree 4 or more with no rational roots is left as it is.

What if a polynomial can’t be factored?

Some can’t be factored with rational numbers — x² + x + 1 has no rational roots, for instance. The calculator says so instead of forcing an answer.

Is factorising the same as factoring?

Yes. “Factorise” (or factorisation) is the UK word and “factor” (or factorization) the US word for the same thing.

Does the factoring calculator show steps?

Yes: the common factor, the x method or the possible rational roots, each factor it divides out, and a final check that multiplies everything back together.

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