Algebra II
Polynomial Long Division
Lesson
Polynomial long division works just like long division of numbers — divide, multiply, subtract, bring down — but with polynomial terms.
Step pattern:
- Divide the leading term of the dividend by the leading term of the divisor. Write that on top.
- Multiply that quotient term by the entire divisor.
- Subtract the result from the dividend.
- Bring down the next term and repeat.
- Stop when the remaining polynomial has degree less than the divisor.
Worked example 1 — clean division (no remainder)
x² ÷ x = x. So x is the first term of the quotient. Multiply x by (x + 2) to get x² + 2x. Subtract: (x² + 5x) − (x² + 2x) = 3x.
Bring down +6. Now we have 3x + 6. Divide 3x ÷ x = 3. Multiply 3 by (x + 2) to get 3x + 6. Subtract: 0. Done.
Worked example 2 — with a remainder
x² ÷ x = x. Multiply: x² + x. Subtract: 3x. Bring down: 3x + 7. Then 3x ÷ x = 3. Multiply: 3x + 3. Subtract: 4.
How to type your answer
Each problem here divides cleanly (no remainder). Type the quotient only, fully simplified. Use ^ for exponents, no spaces, descending order. Examples: x+3, x^2-2x+1, 2x+5.
Practice
Work through these. Stuck? Click Get a hint.
Warm-Up
Quick problems to get going.
Problem 1
Problem 2
Problem 3
Problem 4
Practice
Standard problems matching the lesson.
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Challenge
Harder problems — edge cases, trickier numbers, multiple steps.
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22