Polynomial and Synthetic Division

The Central Idea

Ordinary division can be written in the form

Polynomial division uses exactly the same structure.

If \(P(x)\) is divided by \(D(x)\), then

where:

• \(P(x)\) is the dividend;
• \(D(x)\) is the divisor;
• \(Q(x)\) is the quotient;
• \(R(x)\) is the remainder.

The remainder must have degree less than the degree of the divisor.

Division as the Reverse of Multiplication

Suppose

Then dividing

by

must return

Polynomial division therefore reverses a known multiplication structure.

Exact Division

Division is exact when the remainder is zero.

For example,

is exact because

Therefore,

Division with a Remainder

Not every polynomial divides evenly.

For example, dividing

by

produces quotient

and remainder

Thus

Quotient-Remainder Form

The result of polynomial division can be written in two equivalent ways.

If

then dividing both sides by \(D(x)\) gives

This form is especially useful when working with rational expressions.

Polynomial Long Division

Polynomial long division follows the same repeated pattern as long division with numbers.

At each stage:

1. divide the leading term of the current dividend by the leading term of the divisor;
2. place that result in the quotient;
3. multiply the divisor by the new quotient term;
4. subtract;
5. repeat.

Example: Divide a Quadratic by a Linear Polynomial

Divide

by

Start with the leading terms:

Therefore, the first quotient term is

Multiply and Subtract

Multiply the divisor by \(x\):

Subtract:

This leaves

Repeat the Process

Divide the new leading term by the leading term of the divisor:

Add \(3\) to the quotient.

Multiply:

Subtract:

Therefore,

Long Division Written Compactly

The compact written algorithm is useful, but the meaning remains the same: divide, multiply, subtract, repeat.

Example with a Remainder

Divide

by

First,

Multiply:

Subtract:

Next,

Multiply:

Subtract:

Therefore,

Writing the Same Result as a Rational Expression

From

we obtain

Missing Powers Must Be Included

Polynomial long division depends on aligning like powers.

Consider

Written with missing powers included,

The zero terms act as placeholders and keep the columns aligned during division.

Example with Missing Powers

Divide

by

Write the dividend as

Begin:

Multiply:

Subtract:

Continue the Missing-Power Example

Next,

Multiply:

Subtract:

Then

Multiply:

Subtract:

Therefore,

Degree Predicts the Quotient Degree

When dividing nonzero polynomials,

whenever the dividend degree is at least the divisor degree.

For example, dividing a degree-\(5\) polynomial by a degree-\(2\) polynomial produces a quotient of degree

This gives a useful structural check before the division begins.

When the Divisor Has Higher Degree

If

then the quotient is zero and the entire dividend is the remainder.

For example,

has

and

This satisfies the requirement that

Dividing by a Higher-Degree Polynomial

Long division works even when the divisor is not linear.

Divide

by

Begin with

Multiply:

Subtract:

Continue the Higher-Degree Example

Next,

Multiply:

Subtract:

Then

Multiply:

Subtract:

Since

the division stops.

Result of the Higher-Degree Division

The quotient is

and the remainder is

Therefore,

The Remainder Must Be Smaller Than the Divisor

Long division stops when the remainder has lower degree than the divisor.

If the divisor has degree \(2\), then the remainder may have degree \(1\) or \(0\), but not \(2\) or greater.

This is analogous to ordinary integer division, where the remainder must be smaller than the divisor.

Checking Long Division

Every division result can be checked using

Suppose division gives

and

when dividing \(x^2+4x+7\) by \(x+2\).

Check:

The original dividend returns.

Synthetic Division

Synthetic division is a compact form of polynomial division that works especially well when the divisor has the form

Instead of repeatedly writing variables and powers, synthetic division works directly with coefficients.

Why Synthetic Division Is Possible

When dividing by

the leading coefficient of the divisor is \(1\), and the same sequence of multiply-and-add operations repeats at every stage.

Because the powers are already determined by position, we can suppress the written variables and operate on the coefficient list alone.

Use the Opposite Sign from the Divisor

For a divisor

synthetic division uses the number

Therefore:

• for \(x-3\), use \(3\);
• for \(x+4=x-(-4)\), use \(-4\);
• for \(x-1\), use \(1\).

This sign convention is one of the most common sources of synthetic division errors.

Example: Synthetic Division

Divide

by

The coefficients are

Since the divisor is \(x-1\), use

Synthetic Division Procedure

Start with

Bring down the first coefficient:

Multiply by the synthetic number:

Add to the next coefficient:

Continue the Synthetic Division

Multiply:

Add:

Multiply:

Add:

The synthetic row is

Interpreting the Synthetic Row

The final entry is the remainder:

The preceding entries are the coefficients of the quotient.

Since the original dividend has degree \(3\), the quotient has degree \(2\):

Therefore,

Synthetic Division with a Remainder

Divide

by

Use \(2\) and the coefficients

Bring down \(2\).

Multiply:

Add:

Continue the Remainder Example

Multiply:

Add:

Multiply:

Add:

The synthetic result is

Interpret the Quotient and Remainder

The quotient coefficients are

so

The remainder is

Therefore,

Synthetic Division with \(x+3\)

Divide

by

Rewrite the divisor as

Therefore, use

The coefficients are

Carry Out the \(x+3\) Division

Bring down \(1\).

Multiply:

Add:

Multiply:

Add:

Multiply:

Add:

Therefore,

Missing Powers in Synthetic Division

Missing powers must appear as zero coefficients.

Consider

The complete coefficient list is

Omitting either zero would shift every later coefficient into the wrong power position.

Example with Missing Powers

Divide

by

Use

with coefficients

Bring down \(1\).

Multiply and add:

Interpret the Missing-Power Example

The synthetic row is

Therefore,

and

So

Synthetic Division as Repeated Multiply-and-Add

The synthetic algorithm has a simple rhythm:

1. bring down;
2. multiply by \(c\);
3. add;
4. multiply by \(c\);
5. add;
6. continue until the final coefficient.

The final number is the remainder.

All earlier numbers form the quotient coefficients.

Why the Quotient Degree Drops by One

Synthetic division uses a linear divisor:

A linear divisor has degree \(1\).

Therefore, dividing a degree-\(n\) polynomial by \(x-c\) produces a quotient of degree

If the dividend is cubic, the quotient is quadratic. If the dividend is degree \(7\), the quotient is degree \(6\).

When Synthetic Division Applies

In its standard elementary form, synthetic division applies directly when the divisor is

Examples include

and

For general polynomial divisors, long division remains the universal method.

What About a Divisor Like \(2x-3\)?

The divisor

is linear, but it is not in the simple form \(x-c\) because its leading coefficient is \(2\).

Polynomial long division handles it directly.

More advanced synthetic methods can be adapted to divisors of the form \(ax-b\), but the standard synthetic algorithm in this course will use monic linear divisors \(x-c\).

Long Division and Synthetic Division Produce the Same Result

Synthetic division is not a different kind of division.

It is a compressed bookkeeping system for the same quotient-remainder process.

If both methods are applicable and performed correctly, they must produce the same quotient and remainder.

Comparing the Two Methods

Choosing the Best Method

Use long division when:

• the divisor has degree greater than \(1\);
• the divisor is not in the form \(x-c\);
• you want to see the full polynomial bookkeeping.

Use synthetic division when:

• the divisor is \(x-c\);
• the polynomial has many terms;
• efficient coefficient arithmetic is useful.

Division and Factorization

If division by a polynomial produces remainder zero, then the divisor divides the polynomial exactly.

For example,

Therefore,

This connects division directly to factorization.

The next lesson will formalize this relationship through the Remainder Theorem and Factor Theorem.

Division and Function Evaluation

The final synthetic entry has an important relationship to evaluating the polynomial at the synthetic number.

For example, when dividing by

the final synthetic value matches the value obtained from

We will prove and formalize this fact in the next lesson.

Common Error: Forgetting Missing Terms

Suppose the dividend is

The coefficient list is not

The correct list is

The zeros represent the missing \(x^3\) and \(x^2\) terms.

Common Error: Using the Wrong Synthetic Number

For the divisor

the synthetic number is

because

Using \(5\) would correspond to dividing by \(x-5\), which is a different polynomial.

Common Error: Forgetting the Remainder

Suppose division gives quotient

and remainder

The result is not merely

The complete relationship is

Or, in rational-expression form,

Common Error: Stopping Long Division Too Soon

Long division continues until the remainder has degree less than the divisor.

If the divisor is linear and the current remainder still contains an \(x\)-term, division must continue.

For a linear divisor, the final remainder must be a constant.

Common Error: Misreading the Synthetic Quotient

Suppose synthetic division of a cubic polynomial produces

The last number is the remainder.

The quotient coefficients are

Since the quotient must be quadratic,

The coefficient positions determine the powers.

A Reliable Procedure for Polynomial Long Division

1. Write the dividend and divisor in standard form.
2. Insert zero terms for missing powers.
3. Divide the leading term of the current dividend by the leading term of the divisor.
4. Place the result in the quotient.
5. Multiply the entire divisor by that quotient term.
6. Subtract carefully.
7. Bring down the next term or continue with the remaining polynomial.
8. Repeat until the remainder degree is less than the divisor degree.
9. Verify using \(P=DQ+R\).

A Reliable Procedure for Synthetic Division

1. Confirm that the divisor is in the form \(x-c\).
2. Identify the synthetic number \(c\).
3. Write every coefficient, including zeros for missing powers.
4. Bring down the first coefficient.
5. Multiply by \(c\).
6. Add to the next coefficient.
7. Repeat multiply-and-add through the final coefficient.
8. Interpret the final number as the remainder.
9. Interpret the earlier numbers as quotient coefficients.
10. Check using \(P=DQ+R\).

Example: Full Synthetic Workflow

Divide

by

Use

Coefficients:

Bring down:

Multiply and add:

Therefore,

Example: Full Long-Division Workflow

Divide

by

First:

Multiply:

Subtract:

Next:

Multiply:

Subtract:

Since the remainder has degree \(1\), which is less than the divisor's degree \(2\), stop.

Therefore,

Practice 1: Exact Long Division

Divide:

Practice 2: Long Division with Remainder

Divide:

Practice 3: Missing Powers

Divide:

Practice 4: Higher-Degree Divisor

Divide:

Practice 5: Synthetic Division

Use synthetic division:

Practice 6: Synthetic Division with \(x+2\)

Use synthetic division:

Practice 7: Missing Coefficients

Use synthetic division:

Practice 8: Identify the Quotient Degree

Before dividing, determine the expected quotient degree when a degree-\(8\) polynomial is divided by a degree-\(3\) polynomial.

Practice 9: Identify the Synthetic Number

State the synthetic number used for each divisor:

1. \(x-7\)
2. \(x+3\)
3. \(x-\frac12\)

Practice 10: Verify a Division Result

Verify that

Practice Answers

Practice 1

Therefore,

Practice 2

Divide:

Subtract \(x^2+x\), leaving

Then

Subtract \(2x+2\), leaving

Therefore,

Practice 3

Include the missing term:

The quotient is

with remainder \(0\).

Therefore,

Practice 4

The quotient is

and the remainder is

Therefore,

Practice 5

Use \(2\) with coefficients

Synthetic division gives

Therefore,

Practice 6

Use \(-2\) with coefficients

Synthetic division gives

Therefore,

Practice 7

Use the complete coefficient list

Using \(1\), synthetic division gives

Therefore,

Practice 8

Practice 9

1. \(x-7\): \(\boxed{7}\)
2. \(x+3=x-(-3)\): \(\boxed{-3}\)
3. \(x-\frac12\): \(\boxed{\frac12}\)

Practice 10

Multiply:

Expand:

Add the remainder:

The original dividend is recovered.

What to Remember

Polynomial division follows the quotient-remainder relationship

Polynomial long division is the general algorithm:

• divide leading terms;
• multiply;
• subtract;
• repeat until the remainder has lower degree than the divisor.

Synthetic division is a compressed coefficient method for divisors of the form

Missing powers must always be represented by zero coefficients.

A zero remainder means the division is exact and the divisor is a factor of the polynomial.

Where We Go Next

Synthetic division reveals a striking connection between division and function evaluation: the remainder produced by dividing \(P(x)\) by \(x-c\) is exactly \(P(c)\).

The next lesson, Remainder and Factor Theorems, proves and applies this relationship to evaluate polynomials efficiently, test possible factors, and connect zeros directly to linear factors.