Polynomial Long Division Calculator

Six boxes, one click. Type the coefficients of your dividend and divisor, press Calculate, and this polynomial long division calculator returns the quotient and the remainder at once. The polynomial long division calculator form is pre-filled with the sample below. Press Calculate on those and the polynomial long division calculator answers instantly.

• Dividend: x³ (0 if quadratic) — the number on $x^3$. Type 0 when your dividend has no cubic term. For $x^2+3x+5$, this box of the polynomial long division calculator holds 0.
• Dividend: coefficient of x² — the number on $x^2$, here 1. The polynomial long division calculator reads it as the second box. The polynomial long division calculator labels every box, so nothing hides.
• Dividend: coefficient of x — the number on $x$, here 3.
• Dividend: constant term — the plain number at the end, here 5.
• Divisor: x (in e·x + f) — the number on $x$ in the divisor, here 1.
• Divisor: constant (e·x + f) — the plain number in the divisor, here 1 — the last of the six the polynomial long division calculator needs.

Five output rows come back. The first is a check: it reads true when the divisor is valid, so false means you typed $e = 0$ — the polynomial long division calculator refuses to divide by zero. Three numbers then build the quotient, $p\cdot x^2 + q\cdot x + r$. The last row is the remainder R. For the sample numbers above the polynomial long division calculator reads $p = 0$, $q = 1$, $r = 2$, $R = 3$. So the quotient is $x + 2$ and the remainder is 3. The polynomial long division calculator prints all five rows at once.

The form covers every dividend up to a cubic divided by a linear divisor $e\cdot x + f$. Need a fourth-degree dividend, or a quadratic divisor? The long division polynomials calculator form stops there. The method below still applies to those shapes, and the AI box discusses them with you.

This page is more than a static polynomial long division calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "Divide $2x^3 - 3x^2 + 4x + 5$ by $x + 2$." The AI fills the six boxes and runs the polynomial long division calculator on this page for you. Any wording works — the polynomial long division calculator only needs the six numbers.
• Take a photo of your homework. The AI reads the polynomials and fills the polynomial long division calculator form for you. Photos save typing in the polynomial long division calculator form.
• Ask about the ideas, not just the answer. "Why do we change the signs when subtracting?" The AI explains each move of the polynomial long division calculator scaffold — every number still comes from the polynomial division calculator on this page.
• Work backwards, then keep asking. Give a quotient and a remainder and ask what the dividend was. Follow-ups are welcome on this polynomial long division solver page, so keep asking. The polynomial long division calculator form answers either direction.

Long division of polynomials copies the arithmetic you already know. Divide 178 by 3 and you loop through four moves: divide, multiply, subtract, bring down. You land on 59 with remainder 1, because $178 = 3 \times 59 + 1$. Long division with polynomials runs the identical loop — the columns just hold powers of $x$ instead of place values. That is why variable long division feels familiar on day one, and why the polynomial long division calculator only needs six numbers to describe the whole job. Feed it those six, and the polynomial long division calculator settles the whole division.

Every division statement has the same shape: the dividend equals the divisor times the quotient, plus the remainder. Every polynomial long division calculator answer fills in that sentence.

$x^2 + 3x + 5 = (x + 1)(x + 2) + 3$

Read the picture. The big rectangle is the part the divisor $x + 1$ measures off: width $x + 1$, height $x + 2$. Its four boxes total $x^2 + 3x + 2$. The leftover strip of area 3 is the remainder, too short in degree to extend the height again. The picture is the polynomial long division calculator answer drawn as geometry. Dividing polynomials by long division asks how many times $e\cdot x + f$ fits — exactly like asking how many threes fit in 178. That single question is what the polynomial long division calculator answers numerically.

A dividing polynomials calculator answers that question with numbers — the polynomial long division calculator rows are that answer, split into parts. A dividing two polynomials calculator that shows its work teaches the question too. This polynomial long division calculator page does both, and the sections below walk the full vertical scaffold that the polynomial long division calculator spares you. Run the polynomial long division calculator form first; the scaffold then becomes a read-along. The polynomial long division calculator is the fast lane; the scaffold is the lesson.

Why six number boxes instead of one big expression? Because a polynomial is fully described by its coefficient list. The dividend $a\cdot x^3 + b\cdot x^2 + c\cdot x + d$ and the divisor $e\cdot x + f$ use exactly six numbers, so this dividing polynomials by long division calculator asks for exactly six. Six boxes, one answer set — the polynomial long division calculator keeps it that simple.

• A quadratic dividend? Type 0 for a — $x^2 + 3x + 5$ is really $0x^3 + 1x^2 + 3x + 5$. The polynomial long division calculator treats the leading 0 exactly like any other coefficient. Zero is a real coefficient to the polynomial long division calculator.
• A missing middle works the same way: $x^3 - 3x + 2$ has $b = 0$. Zero coefficients keep every column in place inside the polynomial long division calculator.
• The divisor must stay linear here. Dividing by $x^2 + x + 2$ needs the general method below, which the worked examples make plain. For linear divisors, the polynomial long division calculator form is enough.

The engine behind the polynomial long division calculator form untangles the coefficients directly. With $e = 1$ the arithmetic is clean bookkeeping. With $e = 2$ or more, every quotient term still comes from one leading-term division at a time. What this polynomial division calculator computes is exactly what the vertical method produces — the same quotient and remainder, reached without the scaffold. That is the polynomial long division calculator trade: less writing, identical math. The polynomial long division calculator rows even come out in scaffold order.

Every polynomial long division calculator stands on one identity, the Division Algorithm. Divide a dividend $f(x)$ by a divisor $d(x)$ and two unique pieces come out: a quotient $q(x)$, and a remainder $r(x)$. The remainder is zero, or lower in degree than the divisor. That one line is everything a polynomial long division calculator promises.

$f(x) = d(x)\cdot q(x) + r(x)$

Flip the identity around and you get the self-check every division deserves: multiply the quotient by the divisor, add the remainder, and the dividend must reappear. Arithmetic gave $178 = (3 \times 59) + 1$. Our sample gives $(x + 1)(x + 2) + 3 = x^2 + 3x + 5$. The polynomial long division calculator rows carry the same identity. The picture shows both bars matching at $x = 3$: 20 + 3 = 23. The polynomial long division calculator makes the same check one click away.

When $R = 0$ the divisor divides evenly and becomes a factor of the dividend — the bridge from division to factoring. Run the polynomial long division calculator, multiply back, compare. When the two sides match, your polynomial long division calculator session is done. The polynomial long division calculator check row settles it instantly. One run of the polynomial long division calculator settles every question on this page.

To divide long division polynomials cleanly, every power of $x$ needs its own column. Then one small loop does all the work — it is how to do polynomial long division in seven moves. The polynomial long division calculator automates them; you should still own them:

1. Arrange both polynomials in descending powers of $x$, inserting a 0 coefficient for any missing term.
2. Divide the leading term of the dividend by the leading term of the divisor. That is the next quotient term.
3. Multiply the whole divisor by that new term.
4. Subtract the product from the current row — change every sign, then add.
5. Bring down the next term of the dividend.
6. Repeat steps 2 to 5 until no term is left to bring down.
7. Stop when the remainder is lower in degree than the divisor, and write it as $R$ or as a fraction over the divisor. That stop condition is the last row the polynomial long division calculator prints.

Divide, multiply, subtract, bring down — the same four words that run $178 \div 3$. Each pass adds one term to the quotient, so a cubic dividend takes at most three passes. The polynomial long division calculator runs those passes on coefficients alone. The next three examples run the loop by hand, and the polynomial long division calculator rows sit beside them for checking. Trust the polynomial long division calculator for the arithmetic; own the loop for the exam. The polynomial long division calculator sits up top; the loop lives in the examples below.

Divide $x^2 + 3x + 5$ by $x + 1$ — the exact numbers sitting in the polynomial long division calculator form on this page. The polynomial long division calculator placeholders are exactly these six.

$\begin{array}{r} x+2\phantom{+5} \\ x+1\;\big)\;\underline{\;x^2+3x+5\;} \\ \underline{\;x^2+x\phantom{+5}\;} \\ 2x+5 \\ \underline{\;2x+2\;} \\ 3 \end{array}$

1. Leading terms: $x^2 \div x = x$, so the quotient starts with $x$.
2. Multiply back: $x(x+1) = x^2 + x$. Subtract, bring down: $2x + 5$.
3. Again: $2x \div x = 2$. Multiply: $2(x+1) = 2x + 2$. Subtract: $3$.
4. Degree 0 is below degree 1, so 3 is the final remainder. Six numbers in, four numbers out — the polynomial long division calculator economy.

The quotient is $x + 2$ with remainder 3, so $x^2 + 3x + 5 = (x+1)(x+2) + 3$. Check it by multiplying back — the identity holds exactly. Type 0, 1, 3, 5, 1, 1 into the polynomial long division calculator above and the same four numbers come back: $p = 0$, $q = 1$, $r = 2$, $R = 3$. This polynomial long division calculator pass is the one the area-model picture draws. The polynomial long division calculator and the scaffold agree term by term. That agreement is the polynomial long division calculator doing its one job.

Divide $x^3 - 3x + 2$ by $x + 2$. The dividend skips $x^2$, so write it as $x^3 + 0x^2 - 3x + 2$ before you start — that 0 keeps every column honest. The polynomial long division calculator takes the same dividend with $b = 0$. Missing terms are where hand work drifts and the polynomial long division calculator does not.

$\begin{array}{r} x^2-2x+1\phantom{+2} \\ x+2\;\big)\;\underline{\;x^3+0x^2-3x+2\;} \\ \underline{\;x^3+2x^2\phantom{-3x+2}\;} \\ -2x^2-3x\phantom{+2} \\ \underline{\;-2x^2-4x\phantom{+2}\;} \\ x+2 \\ \underline{\;x+2\;} \\ 0 \end{array}$

1. $x^3 \div x = x^2$. Multiply, subtract, bring down: $-2x^2 - 3x$.
2. $-2x^2 \div x = -2x$. Multiply, subtract, bring down: $x + 2$.
3. $x \div x = 1$. Multiply, subtract: $0$.

The remainder is 0, so $x + 2$ divides evenly: $x^3 - 3x + 2 = (x+2)(x^2 - 2x + 1)$. The quotient even factors as $(x-1)^2$. A zero remainder always means the divisor is a factor of the dividend. Run 1, 0, −3, 2, 1, 2 through the polynomial long division calculator and $R$ comes back 0 — the polynomial long division calculator signature of a factor. When a factor is the goal, the polynomial long division calculator is the fastest test you own. The polynomial long division calculator R-row is that test in one click.

A student's notebook records a finished division: divisor $x + 2$, quotient $2x^2 - 7x + 18$, remainder $-31$. The dividend is smudged out. Rebuild it with the Division Algorithm — multiply, then add:

1. Multiply the quotient by the divisor: $(x+2)(2x^2 - 7x + 18) = 2x^3 - 3x^2 + 4x + 36$.
2. Add the remainder: $2x^3 - 3x^2 + 4x + 36 - 31$.
3. The dividend was $2x^3 - 3x^2 + 4x + 5$. The smudged line matches the polynomial long division calculator output for those six coefficients.

This is also the checking move: after any division, quotient times divisor plus remainder must restore the dividend exactly. Long division examples with polynomials deserve exactly this closing line — make it a habit. The polynomial long division calculator confirms it: type 2, −3, 4, 5, 1, 2 and the rows read $p = 2$, $q = −7$, $r = 18$, $R = −31$. The polynomial long division calculator rebuilt the smudged line in reverse. Checking is division run backwards, and the polynomial long division calculator rows make it a habit.

Everyone learning how to long divide polynomials meets these four slips. The polynomial long division calculator page lists them because each one survives into calculus. The polynomial long division calculator catches the arithmetic ones:

1. Forgetting to change the signs when subtracting. Subtraction means flipping every sign of the lower row, then adding. One missed flip poisons the rest of the scaffold — most wrong answers trace back here.
2. Dropping missing terms. $x^3 - 3x + 2$ must be written with the $0x^2$ in place. A missing term is an invisible column, and every later row drifts left. The polynomial long division calculator keeps the column as a typed 0.
3. Dividing by the whole divisor. Each quotient term comes from leading term divided by leading term only. Trying to divide $3x$ by $x + 1$ in one bite has no clean answer — the loop exists for a reason. The polynomial long division calculator runs that loop three times for a cubic.
4. Writing the remainder like a mixed number. Dividing polynomials long division style ends with quotient plus remainder-over-divisor: $x + 2 + \frac{3}{x+1}$. Gluing the fraction on without the plus sign reads as multiplication.

The polynomial division calculator above never drifts columns or drops a sign. The vertical method needs your care for exactly those two habits — the polynomial long division calculator checks them for you. Compare your scaffold against the polynomial long division calculator rows after every problem. The polynomial long division calculator never tires of that comparison.