Polynomial Factoring & Roots Explorer

Factor polynomials up to degree 4, find all roots with step-by-step solutions, and visualize the graph. Enter coefficients or type the polynomial directly.

Degree

Coefficients

x^3

x^2

constant

Expanded Form

Factored Form

Root 1

x = 1

Root 2

x = 2

Root 3

x = 3

Step-by-Step Solution

Reference Guide

The Rational Root Theorem

If $p(x) = a_n x^n + \cdots + a_0$ has a rational root $\frac{p}{q}$ , then $p$ divides $a_0$ and $q$ divides $a_n$ .

This gives a finite list of candidates to test. For example, if $a_0 = 6$ and $a_n = 1$ , the candidates are $\pm 1, \pm 2, \pm 3, \pm 6$ .

Tip Start with the smallest candidates. If $p(r) = 0$ , then $r$ is a root and you can divide out $(x - r)$ .

Synthetic Division

A shortcut for dividing a polynomial by $(x - r)$ . If the remainder is 0, then $r$ is a root and you can factor out $(x - r)$ .

Algorithm layout

Write the coefficients in a row
Bring the first coefficient straight down
Multiply by $r$ , add to the next coefficient, repeat
The last number is the remainder

Each successful division reduces the degree by 1, so you can chain divisions to fully factor the polynomial.

The Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Once synthetic division reduces a polynomial to degree 2, the quadratic formula finds the remaining roots.

The discriminant determines the number of real roots.

$\Delta > 0$ Two distinct real roots

$\Delta = 0$ One repeated root

$\Delta < 0$ No real roots (two complex roots)

Multiplicity and Graph Behavior

The multiplicity of a root tells you how many times the factor appears. It also determines how the graph behaves at that root.

Odd multiplicity

The graph crosses the x-axis at the root. A single root ( $m = 1$ ) crosses cleanly, while a triple root ( $m = 3$ ) flattens before crossing.

Even multiplicity

The graph touches the x-axis and bounces back without crossing. A double root ( $m = 2$ ) creates a visible bounce, while higher even multiplicities flatten the curve more near the root.