Dividing Polynomials: Long & Synthetic Division
When the divisor has more than one term, you need polynomial long division — the same divide-multiply-subtract-bring-down rhythm as long division with numbers.
It looks busy at first, but each round follows the exact same four steps, and the process ends when there is nothing left to bring down.
The four-step cycle
Divide the leading term of the dividend by the leading term of the divisor to get the next quotient term. Multiply that term by the whole divisor and subtract.
Bring down the next term and repeat. Continue until the remaining polynomial has lower degree than the divisor — that leftover is the remainder.
Keep columns aligned
Line up like powers of in columns, and insert a zero placeholder for any missing power so nothing shifts.
Subtraction is where sign errors creep in; change the signs of the row you are subtracting and add instead.
Worked examples
Example 1: a clean division
Divide .
Answer:
Example 2: the stopping rule
When do you stop dividing?
Answer: When the remainder's degree is below the divisor's
Example 3: a missing power and a nonzero remainder
Divide .
Answer:
Try one yourself
Common questions
What are the four steps?
Divide, multiply, subtract, bring down — then repeat until the remainder has lower degree than the divisor.
What if a power is missing?
Insert a zero placeholder for it (like ) so the columns stay aligned and no terms shift.
How do I know when to stop?
Stop when the polynomial left over has a lower degree than the divisor. That polynomial is the remainder.
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