Synthetic Division Practice Test

Practice quotient, remainder, factor, and parameter problems with synthetic division.

Synthetic Division Practice Test

This test has 20 questions

Synthetic division control strip

Compress the polynomial. Keep the coefficient logic visible.

Synthetic division replaces polynomial symbols with an organized coefficient row, but it works only when the setup is exact. The synthetic number comes from solving the divisor equal to zero, missing powers require zero placeholders, and the final bottom entry has a special role: it is the remainder.

Anchor 01Solve the divisor equal to zero to get the synthetic number.

Anchor 02Insert a zero coefficient for every missing power.

Anchor 03The last bottom entry is the remainder.

Anchor 04A zero remainder means the divisor is a factor.

1. Synthetic division is a coefficient algorithm, not a sign shortcut

The method becomes compact because the variable powers are implied by position. That makes setup discipline more important: one wrong synthetic value or one missing zero coefficient shifts the entire calculation.

Convert divisor to a synthetic value

→

Build the complete coefficient row

→

Bring down, multiply, add

Setup

Solve the divisor

x - c = 0 \Rightarrow x = c

Use the root of the linear divisor, not the printed sign by itself.

Row structure

Positions encode powers

Each coefficient slot represents one descending power, so missing powers must still receive a zero.

Output

Last entry is special

All but the final bottom entry become quotient coefficients; the final entry is the remainder.

2. The synthetic number comes from the root of the divisor

The sign printed in the divisor can be misleading if read mechanically. Rewrite the divisor in the form $x - c$ , then use $c$ .

Sign conversion panel

Read the divisor as an equation and solve it equal to zero.

Divisor

x - 5

Equation

x - 5 = 0

Synthetic value

x = 5

Minus in divisor

x - 3

uses synthetic value

3

Plus in divisor

x + 4

uses synthetic value

- 4

General form

x - c

always uses

c

Quick self-checkSubstitute the synthetic value into the divisor; the result should be zero.

(- 4) + 4 = 0

3. Zero placeholders preserve the power map

Because synthetic division hides the variable symbols, coefficient position carries all the exponent information. If one power is missing, a zero coefficient must mark its slot.

x^{4} + 0 x^{3} - 5 x^{2} + 0 x + 6

power

x^{4}

x^{3}

x^{2}

x

const.

coef.

1

0

- 5

0

6

4. The repeating motion is bring down → multiply → add

Unlike polynomial long division, synthetic division compresses each cycle to arithmetic on coefficients. The pattern repeats from left to right until the final column is reached.

Bring down

First coefficient drops unchanged

The leading coefficient becomes the first quotient coefficient.

Multiply

Multiply by synthetic value

Take the newest bottom entry and multiply it by the synthetic number.

Add

Add within the next column

The sum becomes the next bottom entry, and the cycle repeats.

1 \to 2 \to (- 1) \to 4

5. Separate quotient coefficients from the remainder

The bottom row has two jobs. Every entry except the last describes the quotient; the final entry is the remainder.

Bottom-row split

1

2

- 1

4

The first three values are quotient coefficients. The final value $4$ is the remainder.

Restore quotient powers

If a cubic dividend is divided by a linear divisor, the quotient is quadratic. Thus the quotient row $1$ , $2$ , $- 1$ becomes $x^{2} + 2 x - 1$ .

6. The final entry also answers a factor question

Synthetic division connects directly to the Factor Theorem. A zero final entry means zero remainder, so the divisor is an exact factor of the dividend.

f (c) = 0

Zero remainder

R = 0

The divisor is a factor.

Nonzero remainder

R \neq 0

The divisor is not a factor.

Remainder Theorem link

R = f (c)

The final synthetic entry must agree with direct evaluation.

7. Parameter problems turn a synthetic row into an equation

If a coefficient contains an unknown parameter, the synthetic process may produce a final entry involving that parameter. A condition such as “the divisor is a factor” then gives an equation to solve.

Factor condition

If the last synthetic entry is $k - 7$ and the divisor is known to be a factor, set $k - 7 = 0$ . Therefore $k = 7$ .

Known remainder condition

If the remainder is specified as $5$ , set the final synthetic expression equal to $5$ rather than zero.

8. Reconstruct the dividend to verify the synthetic result

A correct synthetic quotient and remainder must satisfy the polynomial division identity. This is the strongest way to catch a shifted coefficient row or a wrong synthetic sign.

P (x) = (x - c) Q (x) + R

Divisor × quotient

→

Add remainder

→

Recover dividend

9. Error analysis: synthetic division fails at setup first

Most wrong answers can be traced to one of two setup errors: using the wrong synthetic value or omitting a zero coefficient. The remaining mistakes usually occur when interpreting the bottom row.

Printed sign copied directly

For $x + 4$ , the synthetic value is $- 4$ , not $4$ .

Missing power omitted

Skipping a zero coefficient shifts every later position and changes the quotient.

First coefficient not brought down

The leading coefficient drops directly before any multiplication begins.

Multiplied by the wrong value

Every product uses the same synthetic number determined from the divisor.

Last entry treated as quotient coefficient

The final bottom entry is the remainder, not another quotient coefficient.

Zero remainder ignored

A final zero is not merely arithmetic; it confirms that the divisor is a factor.

10. Worked mini-set: interpret the strip correctly

These examples are illustrative teaching examples, not questions copied from the test.

Example A

Choose the synthetic value

Divisor $x - 6$ uses synthetic value $6$ .

Example B

Reverse a plus sign

Divisor $x + 2$ uses synthetic value $- 2$ .

Example C

Missing cubic term

$x^{4} + 0 x^{3} - 3 x^{2} + 1$ must include the zero cubic coefficient.

Example D

Read the remainder

If the final bottom entry is $- 5$ , the remainder is $- 5$ .

Example E

Factor decision

If the final entry is $0$ , the divisor is a factor of the dividend.

Example F

Parameter equation

If the final entry is $k + 3$ and the divisor must be a factor, solve $k + 3 = 0$ .

Final synthetic-division checklist

Before accepting a quotient or factor conclusion, audit the synthetic value, coefficient positions, and interpretation of the last entry.

Did I solve the divisor equal to zero?Use the divisor root as the synthetic number.

Are all powers represented?Insert zero coefficients for every missing power.

Did I bring down the first coefficient unchanged?That starts the bottom row.

Did I repeat multiply then add across every column?The same synthetic value is used each time.

Did I separate quotient entries from the remainder?The final bottom entry is not part of the quotient coefficient list.

Does the factor or reconstruction check agree?Zero remainder means factor; divisor × quotient + remainder must recover the dividend.