Polynomials

What is a polynomial?

A polynomial in x is any expression built from whole-number powers of x, each multiplied by a coefficient and added together:

The highest power n is called the degree. A degree-1 polynomial set to zero is a linear equation; degree 2 is quadratic; degree 3 is cubic. A polynomial of degree n has at most n real roots.

Why synthetic division matters

Once you've found one root of a polynomial — by trial, by the Rational Root Theorem, or by inspection — you'd like to "divide it out" so you're left with a simpler polynomial of one lower degree, which is easier to solve. Synthetic division is a fast, compact shortcut for exactly that division, when dividing by a factor of the form (x − r).

It uses only the coefficients — no x's written out at all — and a short chain of multiply-and-add steps. Try it yourself in the Synthetic Division tab.

Worked example

Divide x³ − 6x² + 11x − 6 by (x − 1) using synthetic division.

Write down just the coefficients: 1, −6, 11, −6. Bring down the first coefficient, then repeatedly multiply by the root (1) and add to the next coefficient:

The bottom row reads: quotient coefficients 1, −5, 6, and a final remainder of 0.

A remainder of 0 confirms (x − 1) really is a factor, and the quotient is one degree lower:

From here, the quadratic factor (x² − 5x + 6) can be solved directly — by factorization it's (x − 2)(x − 3) — giving the full set of roots: x = 1, 2, 3. This is exactly the shortcut used on the Cubic Equations page.

Try loading these same numbers into the Synthetic Division tab: pick Degree 3, enter coefficients 1, −6, 11, −6, and set r = 1.