Synthetic Division Calculator

Synthetic division is a shortcut for dividing a polynomial $p(x)$ by a linear factor $x - k$. It's faster than long division and produces the same quotient and remainder, just with less writing.

Given $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0$ divided by $x - k$, synthetic division produces:

$p(x) = (x - k) q(x) + r$

where $q(x)$ is the quotient (degree $n - 1$) and $r$ is the constant remainder.

Key uses:

1. Quick polynomial division when the divisor is a linear $x - k$.
2. Evaluate $p(k)$ — by the Remainder Theorem, $p(k) = r$, so the remainder is exactly the function value.
3. Factor polynomials — if $r = 0$, then $(x - k)$ is a factor and $q(x)$ tells you the cofactor.
4. Find rational roots combined with the Rational Roots Theorem.

Setup

To divide $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0$ by $x - k$:

1. Write the divisor's zero $k$ on the left.
2. List the coefficients of $p(x)$ on the right, including zeros for any missing terms.

Algorithm

1. Bring down the first coefficient ($a_n$) unchanged.
2. Multiply by $k$ and write the result under the next coefficient ($a_{n-1}$).
3. Add the column. Write the sum on the bottom row.
4. Repeat: multiply that sum by $k$, write under the next coefficient, add.
5. Continue until you finish all coefficients.

Reading the Result

The bottom row contains:

• The first $n$ entries: coefficients of the quotient $q(x)$ (in descending order of degree).
• The last entry: the remainder $r$.

Example: $(x^3 - 4x + 5) \div (x - 2)$

Coefficients of $x^3 + 0x^2 - 4x + 5$: $[1, 0, -4, 5]$. Divisor zero: $k = 2$.

 2 |  1   0  -4   5
   |      2   4   0
   |________________
      1   2   0   5

Quotient: $x^2 + 2x + 0 = x^2 + 2x$. Remainder: $5$.

So $x^3 - 4x + 5 = (x - 2)(x^2 + 2x) + 5$.

Connection to the Remainder Theorem

The remainder $r$ in $p(x) = (x - k)q(x) + r$ equals $p(k)$. Setting $x = k$:

$p(k) = (k - k) q(k) + r = r$

So synthetic division is a quick way to evaluate $p(k)$ without plugging in.

Factor Theorem

A corollary: $(x - k)$ is a factor of $p(x)$ iff $p(k) = 0$ iff the synthetic-division remainder is $0$.

• Missing zero placeholders: For $p(x) = x^3 - 4x + 5$, you must include a $0$ for the missing $x^2$ term. Otherwise the columns misalign.
• Sign error on $k$: To divide by $x - 2$, use $k = 2$ (the zero of the divisor). To divide by $x + 3$, use $k = -3$.
• Cannot use directly for $ax - k$ divisors: Synthetic division as taught works for $x - k$ (leading coefficient 1). For $ax - k$, factor out $a$ first or use polynomial long division.
• Forgetting to drop the first coefficient: The first step is always 'bring down $a_n$' — multiply nothing yet.
• Misreading the quotient: The bottom row's first $n$ entries are coefficients, and the degree drops by 1. A degree-4 polynomial divided by $x - k$ gives a degree-3 quotient.