Polynomial Factorization Calculator
Factor quadratics by cross-method and cubics by the factor theorem; apply the remainder theorem for P(k).
Input Data
Results
At a glance:Factorization writes a polynomial as a product of lower-degree factors. Quadratics use the cross-method: split a and c into factor pairs whose cross-products sum to b. The remainder theorem says the remainder of P(x) ÷ (x−k) is P(k); if P(k)=0 then (x−k) is a factor (factor theorem). Cubics are solved by testing an integer root then reducing to a quadratic.
Formula
Cross-method: a=p·s, c=q·r, p·r+q·s=b ⇒ (px+q)(sx+r).
Remainder: P(k) = remainder of P(x) ÷ (x−k) (Horner's method).
Factor theorem: P(k)=0 ⇔ (x−k) divides P(x).
Cubic reduction: P(x) = (x−r)(A x²+B x+C).
$$P(x) = (px+q)(sx+r)$$$$P(k) = \text{remainder of } P(x) \div (x-k)$$$$P(k)=0 \;\Rightarrow\; (x-k)\mid P(x)$$How to Use
- Pick the degree (quadratic / cubic).
- Enter integer coefficients a, b, c (and d for cubic).
- Enter k to apply the remainder theorem and find P(k).
- The tool returns the factored form and the working.
Common quadratic factorizations
| Polynomial | Factorization | Note |
|---|---|---|
| x²−3x+2 | (x−1)(x−2) | Δ=1 |
| x²−5x+6 | (x−2)(x−3) | Δ=1 |
| 2x²+5x+3 | (2x+3)(x+1) | cross-method |
| x²−4 | (x−2)(x+2) | difference of squares |
A real quadratic with Δ<0 cannot be factored over the reals.
Case Studies
x² − 3x + 2 (cross-method)
a=1, c=2; choose q=−1, r=−2 so q·r=2 and p·r+q·s = −2−1 = −3 ✓.
So x²−3x+2 = (x−1)(x−2).
x³ − 6x² + 11x − 6 (root test)
Try r=1: P(1)=1−6+11−6=0 ⇒ (x−1) is a factor.
Synthetic division gives x²−5x+6 = (x−2)(x−3).
So x³−6x²+11x−6 = (x−1)(x−2)(x−3).
FAQ
How do I start cross-multiplication?
List factor pairs of a (p,s) and of c (q,r), then check whether the cross sum p·r+q·s equals b.
When to use the remainder theorem?
To find P(k) quickly, or to test whether (x−k) is a factor: P(k)=0 means it is (factor theorem).
How to factor a cubic?
Test integer roots (divisors of the constant term), factor out (x−r) by synthetic division, then factor the remaining quadratic.
Must coefficients be integers?
This tool handles integer coefficients for cross-method and integer root tests; scale decimals to integers first.
Related Tools
References
Content review: Calculatorism Science Team. Cross-multiplication, synthetic division and the remainder theorem steps have been verified. Results are for reference only.