Polynomial Long Division

Step-by-step solution

1. 1

   Setup

   Divide P(x) = x^3 - 6x^2 + 11x - 6 by D(x) = x - 2. Repeat: divide the leading terms, multiply, subtract — until the degree of the current remainder drops below 1.

   $$\left(x^{3} - 6x^{2} + 11x - 6\right) \div \left(x - 2\right)$$
2. 2

   Division step 1

   Divide the leading term x^3 by the leading term x to get the next quotient term x^2; multiply D(x) by it and subtract from the current remainder.

   $$\begin{aligned} \frac{1}{1} &= x^{2} \\[4pt] &x^{3} - 6x^{2} + 11x - 6 \\ -{}&\left(x^{3} - 2x^{2}\right) \\ ={}&-4x^{2} + 11x - 6 \end{aligned}$$

   • x^3 ÷ x = x^2
   • x^3 - 6x^2 + 11x - 6 − (x^3 - 2x^2) = -4x^2 + 11x - 6
3. 3

   Division step 2

   Divide the leading term -4x^2 by the leading term x to get the next quotient term -4x; multiply D(x) by it and subtract from the current remainder.

   $$\begin{aligned} \frac{-4}{1} &= -4x \\[4pt] &-4x^{2} + 11x - 6 \\ -{}&\left(-4x^{2} + 8x\right) \\ ={}&3x - 6 \end{aligned}$$

   • -4x^2 ÷ x = -4x
   • -4x^2 + 11x - 6 − (-4x^2 + 8x) = 3x - 6
4. 4

   Division step 3

   Divide the leading term 3x by the leading term x to get the next quotient term 3; multiply D(x) by it and subtract from the current remainder.

   $$\begin{aligned} \frac{3}{1} &= 3 \\[4pt] &3x - 6 \\ -{}&\left(3x - 6\right) \\ ={}&0 \end{aligned}$$

   • 3x ÷ x = 3
   • 3x - 6 − (3x - 6) = 0
5. 5

   Result

   Quotient Q(x) = x^2 - 4x + 3, remainder R(x) = 0.

   $$x^{3} - 6x^{2} + 11x - 6 = \left(x^{2} - 4x + 3\right) \cdot \left(x - 2\right) + 0$$
6. 6

   The divisor is a factor

   The remainder is 0, so x - 2 divides x^3 - 6x^2 + 11x - 6 exactly — it is a factor of the dividend.

   $$x^{3} - 6x^{2} + 11x - 6 = \left(x^{2} - 4x + 3\right) \cdot \left(x - 2\right)$$
7. 7

   Remainder Theorem

   The divisor is linear of the form (x − r) with r = 2. By the Remainder Theorem, the remainder equals P(2) — here 0.

   $$R = P\left(2\right) = 0$$