Dividing Polynomials
Dividing a polynomial \(p(x)\) by \(d(x)\) rewrites it as \(p(x) = d(x)\cdot q(x) + r(x)\), where q(x) is the quotient and r(x) is the remainder — the same idea as \(17 = 5\cdot 3 + 2\) for integers. Synthetic division is a fast shortcut for dividing by a linear \(x - a\).
By the end you'll be able to run synthetic division by hand and read off the quotient and remainder from the bottom row.
Predict: what's the remainder of \((x^2-5x+6)\div(x-2)\)? Set c = 2 below and step through to check.
This steps through dividing \(p(x) = x^2-5x+6\) by \((x-c)\): bring down the leading coefficient, then repeat multiply by c and add down the column. The last bottom-row cell — boxed in orange — is always the remainder.
Set up: p(x)'s coefficients sit in the top row, c sits in the badge on the left.
q(x) = x + (?), remainder = ? — step through to reveal.
Long division on polynomials mirrors long division on numbers, and when the divisor is linear, synthetic division strips it down to a fast coefficients-only routine.
Long division cycles divide, multiply, subtract, bring down — just like dividing 748 by 6 by hand. Synthetic division shortcuts this for a linear divisor \(x-a\): write only the coefficients (inserting 0 for any missing power so columns stay aligned), bring down the leading coefficient, then repeat multiply the last bottom-row value by a, add it to the next coefficient. The bottom row's last entry is the remainder; everything before it are the quotient's coefficients.
Division always produces \(p(x) = d(x)\cdot q(x) + r(x)\) with \(\deg r < \deg d\). Unlike addition, subtraction, and multiplication of polynomials — which always stay inside the polynomials (closure) — division can leave a nonzero remainder, so the quotient alone doesn't recover \(p(x)\); you need \(d(x)\cdot q(x) + r(x)\) for that.
Once you suspect a number \(a\) might be a root of \(p(x)\), synthetic division by \((x-a)\) is the fastest way to check: a remainder of 0 confirms \((x-a)\) divides evenly, and hands you the quotient — a lower-degree polynomial — to keep factoring or solving.
Divide \(3x^3 - 5x^2 - 4x + 4\) by \(x + 2\), so \(c = -2\). Coefficients: 3, −5, −4,
4.
Bring down 3. Multiply \(3\cdot(-2) = -6\); add to −5: \(-11\). Multiply
\(-11\cdot(-2) = 22\); add to −4: \(18\). Multiply \(18\cdot(-2) = -36\); add to 4:
−32.
Quotient \(3x^2 - 11x + 18\), remainder −32. Check:
\((x+2)(3x^2-11x+18) - 32 = 3x^3-5x^2-4x+4\). ✓
Divide \(2x^3 + x^2 - 7x + 3\) by \(x - 1\), so \(c = 1\). Coefficients: 2, 1, −7,
3.
Bring down 2. Multiply \(2\cdot 1 = 2\); add to 1: \(3\). Multiply \(3\cdot 1 = 3\);
add to −7: \(-4\). Now finish it: multiply \(-4\cdot 1 = \) ____; add to 3 to get the
remainder ____.
Reveal the answer
\(-4\cdot 1 = -4\); \(3 + (-4) = -1\). Quotient \(2x^2+3x-4\), remainder −1. Set c = 1 on a fresh run of the stepper above and match its bring-down/multiply/add pattern against these numbers.
More info — why synthetic division only works for linear divisors
Synthetic division is long division with the variable stripped out, which only stays trackable when the divisor is degree 1 (so each subtraction step collapses into a single multiply-and-add). For a quadratic or higher divisor, or one with a leading coefficient other than 1, you're back to full long division — see the Purplemath long-division link in Dive deeper for that general case.
Check your understanding
Use synthetic division to divide \(x^3 - 3x^2 + 2x - 5\) by \(x - 1\). What are the quotient and remainder?
To divide \(x^3 - 8\) by \(x - 2\) with synthetic division, what coefficients belong in the top row?
Which of these divisions can you carry out with the synthetic-division shortcut exactly as taught (a linear divisor \(x - a\))?
Synthetic division of \(p(x)\) by \((x-2)\) gives quotient \(x^2+4x+3\) and remainder 0. Which product confirms \(p(x) = x^3+2x^2-5x-6\)?
Recap
- Division rewrites \(p(x) = d(x)\cdot q(x) + r(x)\), with the quotient's remainder having lower degree than the divisor.
- Synthetic division is the coefficients-only shortcut for a linear divisor \(x-a\): bring down, then repeat multiply-by-a-and-add-down.
- Insert 0 for any missing power so the coefficient columns stay aligned.
- The bottom row's last entry is the remainder; the rest are the quotient's coefficients (one degree lower than p(x)).
Dive deeper
- Purplemath — Polynomial long division Walk through the divide-multiply-subtract-bring-down cycle.
- Purplemath — Synthetic division Run the coefficient-only shortcut for linear divisors.
- Paul's Online Notes — Dividing Polynomials Confirm the p(x) = divisor*quotient + remainder form.
Sources
- Polynomial Long and Synthetic Division