Algebra · Step-by-step guide
Polynomial Division
Divide polynomials using long division and synthetic division, and interpret quotients and remainders.
Start with the central idea
Polynomial long division repeats four moves: divide leading terms, multiply, subtract, and bring down. Synthetic division is a shortcut only for divisors of the form .
How it works
Build the method from meaning before memorizing the moves.
These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.
Write every power
Insert zero-coefficient placeholders for missing powers so terms remain aligned.
Divide leading terms
Each quotient term is determined by the leading term of the current dividend divided by the divisor’s leading term.
Use the remainder theorem
When dividing P(x) by x − c, the remainder equals P(c).
Worked examples
Follow the reason for each line, then try to reproduce it without looking.
Divide by .
- ; multiply to get .
- Subtract to get .
- ; subtraction leaves 0.
Divide by .
- Include the missing 0x term.
- The quotient terms are x and 1.
- The final subtraction leaves 2.
Divide by .
- Use synthetic value 2 with coefficients 2, −3, 4, −5.
- Bring down 2; multiply-add repeatedly.
- The final row is 2, 1, 6, 7.
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Omitting placeholders for missing powers.
Using +c instead of c for the divisor x − c.
Stopping before the remainder degree is smaller than the divisor degree.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Polynomial Operations
Reinforce polynomial structure before and alongside division.
Preview the worksheetA strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.
Practice problems with worked solutions
Solve each problem on paper, use the hint only if needed, then compare every step.
Divide by .
Show hint
Use synthetic value −2.
Show worked solution
- Synthetic coefficients 1, 5, 6 produce 1, 3, 0.
- The quotient coefficients are 1 and 3.
- The remainder is 0.
Answer:
Divide by .
Show hint
Use synthetic value −3.
Show worked solution
- Bring down 2; multiply/add to get row .
- The last value is the remainder.
Answer:
Divide by .
Show hint
Use synthetic value 2 and include the zero x-term.
Show worked solution
- Use coefficients .
- Synthetic division gives .
Answer:
Find the remainder when is divided by .
Show hint
Use the Remainder Theorem.
Show worked solution
- Compute .
- .
Answer: Remainder .
Is a factor of ?
Show hint
Evaluate the polynomial at −1.
Show worked solution
- .
- A zero remainder means the divisor is a factor.
Answer: Yes; the quotient is .
Frequently asked questions
Quick answers before you move on.
What should I remember first?
Polynomial long division repeats four moves: divide leading terms, multiply, subtract, and bring down. Synthetic division is a shortcut only for divisors of the form .
How do I know whether I understand this topic?
You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.
What should I do if I keep making the same mistake?
Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.