Polynomial Division: Synthetic and Long Division Guide
Polynomial division is the algebraic procedure of dividing a polynomial $P(x)$ (the dividend) by another polynomial $D(x)$ (the divisor), producing a quotient polynomial $Q(x)$ and a remainder polynomial $R(x)$:
P(x) = D(x) \cdot Q(x) + R(x)
The Remainder Theorem and Factor Theorem
When dividing any polynomial $P(x)$ by a monic linear binomial $(x - c)$:
- 1The Remainder Theorem: The numerical remainder $R$ is identically equal to evaluating the polynomial at $c$:
R = P(c)
- 1The Factor Theorem: $(x - c)$ is a factor of $P(x)$ if and only if $R = P(c) = 0$.
Synthetic Division (Ruffini's Rule)
For a cubic $a_3 x^3 + a_2 x^2 + a_1 x + a_0$ divided by $(x - c)$:
- Bring down the leading coefficient: $q_2 = a_3$.
- Multiply by $c$ and add to the next coefficient: $q_1 = a_2 + c \cdot q_2$.
- Multiply by $c$ and add again: $q_0 = a_1 + c \cdot q_1$.
- The final sum gives the remainder: $R = a_0 + c \cdot q_0$.
Summary Table of Synthetic Division Steps
| Column | Coefficient | Operation | Result |
|---|---|---|---|
| Step 1 ($x^3$) | $a_3$ | Drop down | $q_2 = a_3$ |
| Step 2 ($x^2$) | $a_2$ | $a_2 + c \cdot q_2$ | $q_1$ |
| Step 3 ($x^1$) | $a_1$ | $a_1 + c \cdot q_1$ | $q_0$ |
| Step 4 ($x^0$) | $a_0$ | $a_0 + c \cdot q_0$ | $R$ (Remainder) |