Polynomial Division

Pre-Calculus · Axiom Academy

Long division, the synthetic shortcut for x - c , and the Remainder Theorem — all packed into one identity: dividend = divisor · quotient + remainder. To divide one polynomial by another, repeat a four-beat loop: divide the leading terms, multiply the divisor by that result, subtract , then bring down the next term. Keep going until the leftover has lower degree than the divisor. Watch the loop run on divided by : When the divisor is linear — exactly x - c — you can drop the variables and work with coefficients alone. Synthetic division is the same arithmetic as long division, compressed into a single row of bring down → multiply by c → add . Here is divided by , so c = 2 : Copy the first coefficient straight down — it starts the bottom row. Multiply the number you just wrote by c and place it under the next coefficient. Add down the column to get the next bottom-row entry. Repeat to the end. All but the last entry are the quotient's coefficients; the last entry is the remainder. Synthetic division works only when you divide by x - c . For a quadratic or higher divisor, fall back on long division. 3. The Identity & the Remainder Theorem Every division rearranges into one identity — . When the divisor is x - c , substituting x = c collapses the term to zero, which leaves the whole value of P equal to the remainder. That's the Remainder Theorem : the remainder is just P(c) . Run the long (or synthetic) division all the way down and read the remainder off the last line.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.