How To Do Long Division Polynomial | Clean Steps, Clean Answers

Polynomial long division follows the same rhythm as number long division, just with x-terms instead of digits. If you keep like powers in neat columns, each line has one job and the process stays steady.

What Polynomial Long Division Gives You

Dividing one polynomial by another produces a quotient and, sometimes, a remainder. The stop rule is simple: the remainder’s degree must be lower than the divisor’s degree.

Before You Start: Set Up Clean Columns

A tidy setup prevents most sign and alignment errors.

Write Polynomials In Descending Powers

List terms from highest power to lowest. If a power is missing, write a 0 coefficient placeholder so your columns do not drift.

• Example: 3x⁴ + 0x³ − 5x² + 2x − 7

Match Like Powers When You Write Lines

Every multiply-back line must sit under the matching powers. If you multiply by 7x, your line needs an x² term and an x term in the correct columns.

How To Do Long Division Polynomial Step By Step

Think of long division as a five-move loop. You repeat the same cycle until the degree rule says stop.

Step 1: Divide The Leading Terms

Divide the first term of the current work line by the first term of the divisor. That becomes the next term in your quotient.

Step 2: Multiply Back By The Full Divisor

Multiply the entire divisor by your new quotient term. Write the result under the current work line, lined up by degree.

Step 3: Subtract The Whole Line

Subtract the entire multiply-back polynomial, not just one term. A safe habit is to put parentheses around the bottom line, flip signs, then add.

Step 4: Bring Down The Next Term

Bring down the next term from the original dividend, including any 0 placeholders you wrote in the setup.

Step 5: Stop Using The Degree Rule

When the leading term of your work line has a lower degree than the leading term of the divisor, you’re done. What remains is the remainder.

A Full Worked Problem With A Remainder

Divide 2x³ + 3x² − 5x + 6 by x − 2.

Cycle 1

2x³ ÷ x = 2x². Multiply back: 2x²(x − 2) = 2x³ − 4x². Subtract and bring down −5x to form the next work line.

                2x^2
                _____________
x - 2 )  2x^3 + 3x^2 - 5x + 6
         -(2x^3 - 4x^2)
         ----------------
                 7x^2 - 5x

Cycle 2

7x² ÷ x = 7x. Multiply back: 7x(x − 2) = 7x² − 14x. Subtract and bring down +6.

                2x^2 + 7x
                _____________
x - 2 )  2x^3 + 3x^2 - 5x + 6
         -(2x^3 - 4x^2)
         ----------------
                 7x^2 - 5x + 6
                -(7x^2 - 14x)
                --------------
                       9x + 6

Cycle 3 And Stop

9x ÷ x = 9. Multiply back: 9(x − 2) = 9x − 18. Subtract: (9x + 6) − (9x − 18) = 24. Since 24 has degree 0 and the divisor has degree 1, the degree rule says stop.

                2x^2 + 7x + 9
                _____________
x - 2 )  2x^3 + 3x^2 - 5x + 6
         -(2x^3 - 4x^2)
         ----------------
                 7x^2 - 5x + 6
                -(7x^2 - 14x)
                --------------
                       9x + 6
                      -(9x - 18)
                      ----------
                           24

(2x³ + 3x² − 5x + 6) ÷ (x − 2) = 2x² + 7x + 9 + 24/(x − 2)

Quick Check: Multiply Back And Add The Remainder

To verify, multiply the divisor by the quotient, then add the remainder. You should land back on the original dividend.

• (x − 2)(2x² + 7x + 9) = 2x³ + 3x² − 5x − 18
• Add 24 to get 2x³ + 3x² − 5x + 6

Line Meanings You Can Use While You Work

This table links the written lines to their roles, so you can catch misalignment early.

Part Of The Work	What You Write	What It Does
Dividend Row	All terms in order, with 0 placeholders	Keeps columns aligned by degree
Divisor	Polynomial outside the bracket	Sets the degree rule for stopping
Quotient Line	Terms written across the top	Adds one term per cycle
Leading-Term Division	First term ÷ first term	Picks the next quotient term and its degree
Multiply-Back Row	(New term) × (entire divisor)	Creates a line that cancels the current lead term
Subtract Step	Subtract the full multiply-back polynomial	Builds the next work line
Bring-Down	Next term from the dividend	Restores the next column so the cycle can repeat
Final Form	Quotient + remainder/divisor	Gives the exact value when a remainder stays

How To Write The Final Answer

Most classes accept two formats.

• Quotient plus fraction: q(x) + r(x)/d(x). This is the clean default when a remainder stays.
• Division algorithm form: p(x) = d(x)q(x) + r(x). This form is nice for checks because it shows the rebuild step directly.

If your remainder is 0, you can drop the fraction. In that case the divisor is a factor of the dividend, since p(x) = d(x)q(x) with no leftover term.

Dividing By A Quadratic Or Trinomial Divisor

The steps do not change when the divisor has degree 2 or higher. The only difference is Step 2: the multiply-back line gets longer because you multiply your new quotient term by every term in the divisor.

Here is a compact sketch of what “multiply back by all terms” looks like when the divisor is x² + x − 1 and your new quotient term is 3x:

• 3x(x² + x − 1) = 3x³ + 3x² − 3x

When you write that line under the work line, the x³ term aligns with x³, the x² term aligns with x², and the x term aligns with x. The subtract step stays the same block subtraction you used in the full example.

Synthetic Division As A Shortcut For x − c

If your divisor is x − c and the leading coefficient is 1, synthetic division can save writing. You still rely on the same idea: build the quotient one degree lower than the dividend and track what remains at the end.

Synthetic division is not a replacement for long division. It works for a narrow divisor shape. Long division handles every polynomial divisor, including 2x − 1, x² + 1, and trinomials.

If you are choosing between them on a test, scan the divisor first. If it matches x − c, synthetic division is an option. If it does not, stick with long division and trust the loop.

Remainders And Plug-In Checks

If the divisor is x − c, there is a quick check after you finish: the remainder equals p(c). You can plug c into the original dividend and see whether you land on the same number as your remainder. This does not replace the multiply-back check, but it’s a fast extra signal.

On the worked problem above, the divisor is x − 2, so c = 2. Plug x = 2 into 2x³ + 3x² − 5x + 6 and you get 24, which matches the remainder.

Long Division Of Polynomials With Less Stress

If you want more worked problems to compare your lines against, these two lessons keep the same cycle and notation style.

Khan Academy’s lesson on polynomial division gives practice with feedback.

OpenStax shows textbook-style examples in Dividing Polynomials, including remainder form.

Common Spots Where Errors Sneak In

• Missing degree: Add 0x^k placeholders before you start.
• Wrong first quotient term: Recheck leading-term division and degrees.
• Subtracting one term at a time: Subtract the full polynomial as a block.
• Stopping too soon: If the remainder’s degree is still at least the divisor’s degree, keep going.

A Short Example With A Non-Monic Divisor

When the divisor starts with 2x or 3x, some students freeze because the first division step includes a fraction or a decimal. You can still keep it clean.

Try (6x³ − 5x² + x − 7) ÷ (2x − 1). The first move is 6x³ ÷ 2x = 3x². Multiply back: 3x²(2x − 1) = 6x³ − 3x². Subtract to get −2x² + x − 7, then repeat: (−2x²) ÷ (2x) = −x. That gives the next term and keeps degrees lined up.

The point is simple: divide lead term by lead term, even when the coefficient is not 1. The columns still do the heavy lifting.

Practice Set With Answers

Work these with the same five-move loop. Write placeholders where a degree is missing.

Practice Dividend ÷ Divisor	Quotient	Remainder
(x2 + 5x + 6) ÷ (x + 2)	x + 3	0
(3x2 − 2x − 8) ÷ (x − 2)	3x + 4	0
(2x3 − x + 1) ÷ (x + 1)	2x2 − 2x + 1	0
(x3 + 0x2 − 4x + 4) ÷ (x − 2)	x2 + 2x	4
(4x3 + 3x2 − 10) ÷ (x + 2)	4x2 − 5x + 10	−30
(6x3 − 5x2 + x − 7) ÷ (2x − 1)	3x2 − x	−7
(x4 − 1) ÷ (x − 1)	x3 + x2 + x + 1	0
(2x4 + x2 − 3) ÷ (x2 + 1)	2x2 − 1	−2

Wrap-Up: Keep The Loop, Trust The Columns

Long division is a repeatable loop: divide leading terms, multiply back, subtract, bring down, repeat. With placeholders and straight columns, you can work even longer polynomials without losing the thread.