Dividing Polynomials: Long Division and Synthetic Division

Quick answer

Dividing polynomials works exactly like long division of numbers: divide the leading terms, multiply back, subtract, and bring down the next term. The result is a quotient and a remainder, written a(x)/b(x) = q(x) + r(x)/b(x), exactly as 17/5 = 3 + 2/5. When the divisor is x − a, synthetic division keeps only the coefficients and gets the same answer in a fraction of the writing.

What you'll learn

Divide a polynomial by another using long division
Write a quotient in the form q(x) + r(x)/b(x)
Use synthetic division to divide by x − a

The same algorithm as dividing numbers

Divide $17$ by $5$ and you get $3$ with $2$ left over:

\frac{17}{5} = 3 + \frac{2}{5} \qquad\text{because}\qquad 17 = 5 \times 3 + 2

Polynomials divide the same way, with a quotient and a remainder:

\frac{a(x)}{b(x)} = q(x) + \frac{r(x)}{b(x)} \qquad\text{because}\qquad a(x) = b(x)\,q(x) + r(x)

The numbers stop when the remainder is smaller than the divisor. Polynomials stop when the remainder has smaller degree than the divisor.

Long division, step by step

Divide $2x^3 - 3x^2 + 4x - 5$ by $x - 2$ .

Each round has four moves: divide, multiply, subtract, bring down.

Round	Divide leading terms	Multiply by $x - 2$	Subtract, and what is left
1	$2x^3 \div x = 2x^2$	$2x^3 - 4x^2$	$x^2 + 4x$
2	$x^2 \div x = x$	$x^2 - 2x$	$6x - 5$
3	$6x \div x = 6$	$6x - 12$	$7$

The quotient collects the three answers from the first column: $2x^2 + x + 6$ . The remainder is $7$ , a constant — degree $0$ , less than the divisor’s degree $1$ — so the division stops.

\frac{2x^3 - 3x^2 + 4x - 5}{x - 2} = 2x^2 + x + 6 + \frac{7}{x - 2}

Check by multiplying back. $(x - 2)(2x^2 + x + 6) + 7$ should rebuild the original:

2x^3 + x^2 + 6x - 4x^2 - 2x - 12 + 7 = 2x^3 - 3x^2 + 4x - 5 \quad\checkmark

Missing terms need a zero

Divide $x^3 - 7x + 6$ by $x - 1$ .

There is no $x^2$ term. Write it as $x^3 + 0x^2 - 7x + 6$ before starting.

In long division of numbers, $507$ needs its $0$ in the tens place, or the $5$ and the $7$ land in the wrong columns. A polynomial’s missing power is that same zero, and leaving it out misaligns every step after it.

The division gives $x^2 + x - 6$ with remainder $0$ , so $x - 1$ divides the polynomial exactly.

Synthetic division

When the divisor has the form $x - a$ , every round of long division does the same arithmetic on the coefficients. Synthetic division keeps only that arithmetic.

Divide $2x^3 - 3x^2 + 4x - 5$ by $x - 2$ again. Here $a = 2$ .

Write the coefficients: $2, \; -3, \; 4, \; -5$ .
Bring down the first: $2$ .
Multiply by $a$ and add to the next coefficient: $2 \times 2 = 4$ , and $-3 + 4 = 1$ .
Repeat: $1 \times 2 = 2$ , and $4 + 2 = 6$ . Then $6 \times 2 = 12$ , and $-5 + 12 = 7$ .

	$2$	$-3$	$4$	$-5$
add		$4$	$2$	$12$
result	$2$	$1$	$6$	$7$

The last number, $7$ , is the remainder. The others, $2, 1, 6$ , are the quotient’s coefficients, one degree lower than the original: $2x^2 + x + 6$ .

Use $a$ , not $-a$ . Dividing by $x + 3$ means $a = -3$ , because $x + 3 = x - (-3)$ .

Why synthetic division works

In long division by $x - a$ , each round divides a leading term by $x$ — that only drops the power by one, so the coefficient carries straight down. Then the round multiplies by $x - a$ and subtracts, which is the same as adding $a$ times the coefficient.

Synthetic division does exactly those two things: bring the coefficient down, and add $a$ times it to the next column. Every other symbol in long division is bookkeeping it leaves out.

That is also why it only works for divisors of the form $x - a$ . For a divisor such as $x^2 + 1$ or $2x + 3$ , use long division.

Worked examples

Common mistakes

Practice problems

Divide $x^2 + 5x + 6$ by $x + 2$ .

Answer

$x + 3$

Full solution

Synthetic division with $a = -2$ on $1, 5, 6$ gives $1, 3$ and remainder $0$ .
Divide $x^2 + 3x + 5$ by $x + 1$ .

Answer

$x + 2 + \tfrac{3}{x + 1}$

Full solution

With $a = -1$ : $1, 3, 5$ gives $1, 2$ and remainder $3$ .
Divide $2x^2 + 5x - 3$ by $x + 3$ .

Answer

$2x - 1$

Full solution

With $a = -3$ : $2, 5, -3$ gives $2, -1$ and remainder $0$ .
What value of $a$ do you use to divide by $x - 5$ synthetically?

Answer

$5$

Full solution

The divisor is $x - a$ with $a = 5$ .
Write the coefficient list for $x^3 + 4$ .

Answer

$1, 0, 0, 4$

Full solution

The $x^2$ and $x$ terms are missing, so each gets a $0$ .
Divide $x^3 - 6x^2 + 11x - 6$ by $x - 1$ .

Answer

$x^2 - 5x + 6$

Full solution

With $a = 1$ : $1, -6, 11, -6$ gives $1, -5, 6$ and remainder $0$ .
Divide $x^3 + 2x^2 - 5x - 6$ by $x - 2$ .

Answer

$x^2 + 4x + 3$

Full solution

With $a = 2$ : $1, 2, -5, -6$ gives $1, 4, 3$ and remainder $0$ .
Divide $2x^3 + x^2 - 13x + 6$ by $x - 2$ , then check by multiplying back.

Hint

Synthetic division with $a = 2$ .

Answer

$2x^2 + 5x - 3$

Full solution

Coefficients $2, 1, -13, 6$ . Bring down $2$ . Then $2 \times 2 = 4$ and $1 + 4 = 5$ ; $5 \times 2 = 10$ and $-13 + 10 = -3$ ; $-3 \times 2 = -6$ and $6 - 6 = 0$ .

Quotient $2x^2 + 5x - 3$ , remainder $0$ .

Check: $(x - 2)(2x^2 + 5x - 3) = 2x^3 + 5x^2 - 3x - 4x^2 - 10x + 6 = 2x^3 + x^2 - 13x + 6$ ✓
When dividing by $x^2 + 1$ , what is the largest degree the remainder can have?

Answer

$1$

Full solution

The remainder’s degree must be less than the divisor’s degree, which is $2$ .
Dividing $x^3 - 7x + 6$ by $x - 1$ , Tom uses the coefficients $1, -7, 6$ and gets quotient $x - 6$ with remainder $0$ . Find his error.

Hint

Which power of $x$ is missing?

Answer

He left out the $0$ for $x^2$ . The quotient is $x^2 + x - 6$ .

Full solution

With only three coefficients, synthetic division treats the polynomial as $x^2 - 7x + 6$ — a different polynomial, one degree lower.

The correct list is $1, 0, -7, 6$ . Bring down $1$ ; then $0 + 1 = 1$ ; then $-7 + 1 = -6$ ; then $6 - 6 = 0$ .

Quotient $x^2 + x - 6$ , remainder $0$ .

Checking catches Tom’s version at once: $(x - 1)(x - 6) = x^2 - 7x + 6$ , which is not the cubic he started with.

Frequently asked questions

How does polynomial long division work?

Divide the leading term of what is left by the leading term of the divisor, multiply the whole divisor by that result, subtract, and repeat until what is left has a smaller degree than the divisor.

When do I stop dividing?

When the remainder's degree is less than the divisor's. Dividing by x − 3 stops at a constant remainder.

What is synthetic division?

A shortcut for dividing by x − a that keeps only the coefficients. It gives the same quotient and remainder as long division.

Why do I need a 0 for a missing term?

Every power needs a column. Skipping x² in x³ + 4 would line the other terms up under the wrong powers.

How do I check a division?

Multiply the divisor by the quotient and add the remainder. The result must be the original polynomial.

What to learn next

The Remainder Theorem and the Factor TheoremWhy dividing p(x) by x − a leaves the remainder p(a), why p(a) = 0 means x − a is a factor, and how the two theorems turn one known root into a full factorization.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSA.APR.D.6Arithmetic with Polynomials and Rational ExpressionsRewrite simple rational expressions in different forms; write a(x /b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Frequently asked questions

How does polynomial long division work?

When do I stop dividing?

What is synthetic division?

Why do I need a 0 for a missing term?

How do I check a division?

What to learn next

Standards alignment

More Algebra 2 lessons