Synthetic Division Calculator

Five boxes, one click. Type the four coefficients of your dividend plus the root k. Press Calculate, and this synthetic division calculator returns the whole bottom row of the tableau at once. The synthetic division calculator form is pre-filled with the sample below. Press Calculate on those numbers and the synthetic division calculator answers instantly. No variables, no scaffold — the synthetic division calculator works on numbers alone.

• Dividend: coefficient of x³ — the number on $x^3$. Type 0 when your dividend is only a quadratic. For $2x^3-3x^2+5$ this box of the synthetic division calculator holds 2.
• Dividend: coefficient of x² — the number on $x^2$, here $-3$. The synthetic division calculator labels every box, so nothing hides.
• Dividend: coefficient of x — the number on $x$, here 0. That 0 is the missing-term placeholder: $2x^3-3x^2+5$ has no $x$ term, and the synthetic division calculator keeps its column alive with the 0.
• Dividend: constant term — the plain number at the end, here 5 — the last of the five the synthetic division calculator needs.
• k — the divisor is (x − k) — the root of your divisor. For $x-2$ type 2; for $x+3$ type $-3$. The sign flips at this box, and the synthetic division calculator takes the flipped one.

Four output rows come back from the synthetic division calculator, and they are the tableau's bottom row split into pieces. Row p is the brought-down leading coefficient — the $x^2$ term of the quotient. Rows q and r are the two multiply-add results — the $x$ and constant terms of the quotient. The last row is the remainder. For the sample numbers the synthetic division calculator reads $p = 2$, $q = 1$, $r = 2$, $R = 9$. So the quotient is $2x^2 + x + 2$ and the remainder is 9. The synthetic division calculator prints all four rows at once.

The form divides any dividend up to a cubic by a divisor of the form $x - k$. For plain $x - k$ divisors the synthetic division calculator needs no preparing move at all. Is your divisor $2x - 4$ instead? Factor it as $2(x-2)$, run the synthetic division calculator with $k = 2$, then halve the quotient it reports. The remainder stays untouched. The section on divisors not in $x - k$ form walks that move in full.

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

• Ask in plain words. "Divide $2x^3 - 3x^2 + 5$ by $x - 2$." The AI fills the five boxes and runs the synthetic division calculator on this page for you. Any wording works — the synthetic division calculator only needs the five numbers.
• Take a photo of your homework. The AI reads the polynomials and fills the synthetic division calculator form for you. Photos save typing in the synthetic division calculator form.
• Ask about the ideas, not just the answer. "Why does synthetic division add instead of subtract?" The AI explains each bring-down step — every number still comes from the 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, so keep asking. The synthetic division calculator page keeps teaching.

Synthetic division is polynomial long division with everything non-essential deleted. When the divisor is $x - k$, the variables never change from step to step. Each pass divides leading terms, multiplies back, and subtracts. Because the divisor is linear, that whole scaffold collapses into one small table of coefficients. A synthetic division calculator automates the whole table.

That table is the method. A synthetic division calculator is the table with the arithmetic done for you. This synthetic division calculator asks for five numbers and returns the finished bottom row, cell by cell. The synthetic division calculator never writes a variable — only the five numbers and their multiply-add results.

Every division still closes with the same identity. The dividend equals the divisor times the quotient, plus the remainder. A synthetic division calculator answer is that sentence with numbers in it.

$2x^3 - 3x^2 + 5 = (x - 2)(2x^2 + x + 2) + 9$

The trick behind the shorthand is one sign change. Subtracting a multiple of $x - k$ works out the same as adding a multiple of $k$. So the four moves of long division — divide, multiply, subtract, bring down — shrink to three: bring down, multiply, add. Fewer moves, no variables to copy, same answer. The synthetic division calculator runs all three moves in one click.

Read the picture. The 2 at the far left is $k$, the root of the divisor. Across the top sit the dividend coefficients 2, $-3$, 0, 5, and the 0 marks the missing $x$ term. Each new bottom number comes from multiplying the previous one by $k$ and adding the next column. The bottom row reads 2, 1, 2 — the quotient $2x^2 + x + 2$ — and the final cell is the remainder, 9. That one picture is the entire computation the synthetic division calculator runs on every click. Every synthetic division calculator output row is one cell of that bottom line.

The layout is older than any machine that runs it. Ray's New Higher Algebra, an 1866 textbook, calls synthetic division "an abridgment of the method of division by Detached Coefficients". The same page divides $5x^4 - 12x^3 + 3x^2 + 4x - 5$ by $x - 2$ in one line of figures, landing on the quotient $5x^3 - 2x^2 - x + 2$ with remainder $-1$. One more degree only means one more column. The synthetic division calculator form here covers dividends up to cubics; the AI box happily discusses the longer ones with you. The synthetic division calculator on this page runs the same table that 1866 textbook printed in ink.

To run a synthetic division tableau by hand you prepare two rows, then loop. Write $k$ — the divisor's root — alone at the left of the table. Write the dividend's coefficients across the top, in descending powers, with 0 for every missing term. Then the loop takes over. This is the part a synthetic division calculator automates completely. These are the moves the synthetic division calculator makes on every click:

1. Bring down the leading coefficient. It becomes the first quotient coefficient, unchanged.
2. Multiply that number by $k$. Write the product in the next column, one row down.
3. Add the column. The sum is the next number of the bottom row.
4. Repeat the multiply-and-add for every column that remains.
5. Stop and read. The bottom row, left of the last entry, is the quotient. The last entry is the remainder.

Loop those five lines and you have done everything a synthetic division calculator does. Two reading rules finish the job. The quotient's degree is one less than the dividend's — a cubic dividend gives a quadratic quotient, so 2, 1, 2 reads $2x^2 + x + 2$. The remainder is a constant, because it must sit lower in degree than the linear divisor. The synthetic division calculator rows follow both rules automatically. The loop is short enough to memorize — and the synthetic division calculator runs it error-free.

When the divisor is not in x − k form

The synthetic division calculator takes divisors with leading coefficient 1 — the $x - k$ form. Divisors like $2x - 4$ or $3x + 6$ need one preparing move. Factor out the leading coefficient first: $2x - 4 = 2(x-2)$ and $3x + 6 = 3(x+2)$. Run the synthetic division calculator with the inner root — $k = 2$ or $k = -2$. Put the factored-out number in the leading-coefficient box. The quotient rows come back already divided; the remainder is not divided — it is already final.

Watch the asymmetry, because it trips people. Quotient rows get halved for $2x - 4$; the remainder row does not. Test the rescale: type k = 2 with leading coefficient 2, and the halved quotient comes out of the synthetic division calculator. The worked problem on a $3x - 6$ divisor below shows the full bookkeeping with leading coefficient 3.

The four output rows of the synthetic division calculator are four little formulas — Horner's scheme in the open. Bring down $a$; each later row multiplies the previous result by $k$ and adds the next coefficient:

$p = a$

$q = b + k\cdot p$

$r = c + k\cdot q$

$R = d + k\cdot r$

Type the five sample numbers into the synthetic division calculator and the four rows land on these formulas exactly. The synthetic division calculator is Horner's scheme with a button.

The last formula hides a theorem. Unroll the rows symbolically and $R$ becomes $ak^3 + bk^2 + ck + d$ — exactly $f(k)$, the polynomial's value at the divisor's root. That is the Remainder Theorem: divide $f(x)$ by $x - k$ and the remainder is $f(k)$ — the theorem every synthetic division calculator quietly uses. The picture shows the arithmetic for the sample. The term $2\cdot 2^3 = 16$ starts the line. Subtracting $3\cdot 2^2 = 12$ lands on 4. Adding the constant 5 lands on 9. The remainder 9 and the value $f(2)$ are the same number, and the synthetic division calculator remainder row doubles as a one-shot evaluation.

A zero in that row is even louder. When $R = 0$, the Factor Theorem says $x - k$ is a factor of the dividend. Factor hunting is the number-one use of synthetic division in Algebra 2, and one run of the synthetic division calculator settles it. Multiply back to close the loop: $(x-2)(2x^2+x+2) + 9 = 2x^3 - 3x^2 + 5$, the dividend itself. The synthetic division calculator identity check is that one line. Every synthetic division calculator promise on this page rests on those four formulas.

Divide $2x^3 - 3x^2 + 5$ by $x - 2$ — the exact five numbers sitting in the synthetic division calculator form on this page. The dividend skips its $x$ term, so a 0 takes that column before anything else happens. An 1866 rule says it best: "If any term is wanting, its place must be supplied with a zero."

$\begin{array}{c|rrr|r} 2 & 2 & -3 & 0 & 5\\ \hline & 4 & 2 & 4 & \\ & 2 & 1 & 2 & 9\\ \end{array}$

1. Bring down the 2 — the first quotient coefficient.
2. Multiply $2\cdot 2 = 4$; add the column: $-3 + 4 = 1$.
3. Multiply $1\cdot 2 = 2$; add the column: $0 + 2 = 2$.
4. Multiply $2\cdot 2 = 4$; add the column: $5 + 4 = 9$ — the remainder.

The quotient is $2x^2 + x + 2$ with remainder 9, so $2x^3 - 3x^2 + 5 = (x-2)(2x^2+x+2) + 9$. Multiply back to check — the identity holds exactly. Type 2, $-3$, 0, 5, 2 into the synthetic division calculator above and the same four numbers return: $p = 2$, $q = 1$, $r = 2$, $R = 9$. This pass is the one the tableau picture draws. The hand tableau and the synthetic division calculator agree cell by cell. That agreement is the synthetic division calculator doing its one job.

Divide $4x^3 + 10x^2 - 6x - 20$ by $x + 2$. The divisor reads $x$ plus 2, so $k = -2$ — the sign flips before anything else moves. This example runs the same numbers as OpenStax College Algebra's synthetic division lesson, and the synthetic division calculator takes them as typed. The synthetic division calculator handles the flip inside the $k$ box.

$\begin{array}{c|rrr|r} -2 & 4 & 10 & -6 & -20\\ \hline & -8 & -4 & 20 & \\ & 4 & 2 & -10 & 0\\ \end{array}$

1. Bring down 4. Multiply $4\cdot(-2) = -8$; add: $10 + (-8) = 2$.
2. Multiply $2\cdot(-2) = -4$; add: $-6 + (-4) = -10$.
3. Multiply $-10\cdot(-2) = 20$; add: $-20 + 20 = 0$.

The remainder is 0, so $x + 2$ divides evenly and is a factor: $4x^3 + 10x^2 - 6x - 20 = (x+2)(4x^2 + 2x - 10)$. The quotient even factors as $2(2x^2 + x - 5)$. A zero bottom cell is the Factor Theorem in action. Type 4, 10, $-6$, $-20$, $-2$ into the synthetic division calculator above and $R$ returns 0 — the synthetic division calculator signature of a factor. The synthetic division calculator zero row is the green light. When a factor is the goal, the synthetic division calculator is the fastest test you own.

A student's notebook holds a finished division: divisor $x - 3$, quotient $3x^2 - 5x + 4$, remainder 7. The dividend line is smudged out. Rebuild it with the division identity — multiply, then add:

1. Multiply the quotient by the divisor: $(x-3)(3x^2 - 5x + 4) = 3x^3 - 14x^2 + 19x - 12$.
2. Add the remainder: $3x^3 - 14x^2 + 19x - 12 + 7$.
3. The dividend was $3x^3 - 14x^2 + 19x - 5$.

Now audit the rebuild by running the table forward on the recovered dividend:

$\begin{array}{c|rrr|r} 3 & 3 & -14 & 19 & -5\\ \hline & 9 & -15 & 12 & \\ & 3 & -5 & 4 & 7\\ \end{array}$

Every cell matches the notebook. Type 3, $-14$, 19, $-5$, 3 into the synthetic division calculator above and the rows read $p = 3$, $q = -5$, $r = 4$, $R = 7$ — the smudged line rebuilt and confirmed in one pass. The synthetic division calculator pass audits the rebuild. Checking is division run backwards, and the synthetic division calculator rows make it a habit. The synthetic division calculator never tires of that comparison.

Everyone learning the tableau meets these three slips. The synthetic division calculator page lists them because each one survives into calculus. The synthetic division calculator catches all three:

1. Flipping the sign of k wrong. The divisor $x + 2$ means $k = -2$, not $+2$. Run Example 2 with $k = 2$ by mistake and the last cell reads 40 instead of 0 — the error surfaces at the remainder, every time. The synthetic division calculator takes k exactly as you type it.
2. Skipping the 0 placeholders. A missing term is a missing column. $2x^3 - 3x^2 + 5$ must enter as 2, $-3$, 0, 5, or every later cell drifts. Ray's 1866 rule again: the wanting term's place "must be supplied with a zero." The synthetic division calculator keeps the column as a typed 0.
3. Dividing by the whole divisor, or forgetting the rescale. The tableau never divides — it only multiplies by $k$ and adds. And for a divisor like $3x - 6$, the fix is factoring out the 3 first, then dividing the reported quotient by 3. The remainder never gets divided. Forgetting that last step leaves a quotient three times too large — the synthetic division calculator raw rows make the rescale visible.

Compare your hand tableau against the synthetic division calculator rows after every problem. The synthetic division calculator never flips a sign or drops a column.