How to use it
Type the polynomial in the first box, using ^ for exponents: x^4 - 5x^2 + 4. You can also list just the coefficients in order, separated by spaces: 1 0 -5 0 4. Then choose what you want to do.
- Divide by x − c. Enter c in the second box. Watch the sign: to divide by \(x + 2\), enter \(c = -2\). You get the synthetic division table, the quotient \(Q(x)\) and the remainder \(R\).
- Factor. Leave the second box blank. The tool lists the possible rational zeros, tests them one by one, and every time it finds a zero it divides and carries on with the quotient.
Reading the table
The top row holds the coefficients, with a vertical bar before the constant term, and c sits in the box on the left. Bring the first coefficient straight down. Then multiply the number you just wrote by c, put the product in the next column, and add. The bottom row gives the coefficients of the quotient, which is one degree lower than P(x), and the last entry is the remainder.
The Remainder Theorem gives you a free check: dividing by \(x - c\) leaves a remainder of \(P(c)\). So a remainder of 0 means c is a zero of the polynomial and \(x - c\) is a factor (that's the Factor Theorem).
Where the candidates come from
For a polynomial with integer coefficients, the Rational Root Theorem says every rational zero is \(\pm\frac{p}{q}\), where p divides the constant term and q divides the leading coefficient. For \(2x^3 - 3x^2 - 11x + 6\) the candidates are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm\frac{1}{2}, \pm\frac{3}{2}\), and the ones that work are \(\frac{1}{2}\), \(-2\) and \(3\). That gives \(2x^3 - 3x^2 - 11x + 6 = (2x - 1)(x + 2)(x - 3)\).
If no candidate works, the polynomial has no rational zeros and synthetic division alone won't factor it. When what's left is a quadratic, switch to the quadratic formula calculator.