Polynomial Long Division
Divide polynomials the way you divide numbers.
About this topic
Polynomial long division works like the long division of numbers. Divide the leading term of the dividend by the leading term of the divisor, multiply back, subtract, bring down the next term and repeat until the remainder has lower degree than the divisor.
The result is written as quotient plus remainder over divisor. To divide x² + 5x + 7 by x + 2: x² divided by x gives x; multiply x(x + 2) = x² + 2x; subtract to get 3x + 7; 3x divided by x gives 3; multiply 3(x + 2) = 3x + 6; subtract to leave a remainder of 1. So the answer is x + 3 + 1/(x + 2). If the remainder is zero, the divisor is a factor.
Long division is how you reduce a high-degree polynomial after finding one root, so the rest can be solved with a simpler equation. It is also used to find slant asymptotes of rational functions and in coding theory.
Write the dividend in standard form with placeholders for missing powers, such as 0x², or the columns will not line up. The most common error is subtracting incorrectly, so change the signs of the whole line being subtracted.
Key ideas
- Divide the leading terms
- Multiply back and subtract
- Repeat until the degree is smaller
Quick check
Use cases: where Polynomial Long Division helps
- Homework: review the key idea behind Polynomial Long Division before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Polynomial Long Division without notes.
- Visual learning: play with the interactive lab to see how Polynomial Long Division changes when you adjust its inputs.
- Teaching a friend: explain Polynomial Long Division in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Polynomial Long Division?
Divide polynomials the way you divide numbers.
What will I learn in Polynomial Long Division?
Divide the leading terms. Multiply back and subtract. Repeat until the degree is smaller.
Is Polynomial Long Division part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Polynomial Long Division lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Polynomial Long Division?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.