Synthetic Division Practice Problems With Answers

Synthetic Division Practice Problems

The synthetic division lesson sets out the method; this set drills it. Every problem was built by choosing the divisor, quotient and remainder first and multiplying them out, so the division always comes out exactly as described.

How the difficulty levels work

Difficulty	Dividend	Quotient	What you are asked for
Easy	quadratic	linear	the remainder — the last cell
Medium	cubic	quadratic	the quotient’s constant or coefficient
Hard	quartic	cubic	a middle coefficient of the quotient

Easy asks only for the final cell, which is where most people start: three multiply-and-add steps and read off the answer.

Medium asks for a coefficient inside the quotient instead, so you have to know which bottom-row cells are quotient coefficients and which one is the remainder.

Hard uses a quartic dividend and asks for a coefficient in the middle of the cubic quotient — far enough in that an early arithmetic slip has nowhere to hide.

The layout

For , write in the box and the dividend’s coefficients across the top. Bring the first coefficient straight down, then repeatedly multiply by and add to the next column.

The bottom row gives the quotient’s coefficients, and the final cell is the remainder — which the remainder theorem tells us is .

A worked problem

Divide by .

The root is , and the coefficients are .

Step 1 — bring down the 2.

Step 2 — multiply and add. , and .

Step 3 — again. , and .

Step 4 — last column. , and .

The bottom row reads :

Asked for the remainder you type 9; asked for the quotient’s constant term you type −2; asked for its coefficient you type 3.

Compare that with the same division done the long way — identical answer, roughly a quarter of the writing.

Where people lose marks

Putting the divisor’s sign in the box. means . Flip the sign every time.
Skipping a zero placeholder. needs coefficients .
Counting the last cell as a quotient coefficient. It is the remainder. A cubic divided by a linear factor gives a quadratic quotient — three coefficients, not four.
Adding when you should multiply. The pattern is multiply the running value by , then add down the column. Never the reverse.
Using it on a quadratic divisor. It does not work. Use long division instead.

Frequently Asked Questions

When can I use synthetic division instead of long division?+

Only when the divisor is linear and monic — of the form x − a. For a divisor like 2x − 3, factor the 2 out first and divide the resulting quotient by 2; for any quadratic or higher divisor you need full long division.

Do I write a or negative a in the box?+

You write a, the root — the value that makes the divisor zero. Dividing by x − 3 puts 3 in the box; dividing by x + 3 puts −3 in the box, because x + 3 is x − (−3). This sign is the single most common error in the method.

Why is the quotient one degree lower than the dividend?+

Because you divided by something of degree 1. Dividing a cubic by a linear factor leaves a quadratic, so the bottom row has one fewer coefficient than the top row — the last cell is the remainder, not a coefficient.

What if a degree is missing from the dividend?+

Write a zero for it. Synthetic division works purely on position, so a missing x² term must appear as a 0 in that column or every subsequent value is wrong. This bites hardest on expressions like x³ − 8.

How does this connect to the remainder theorem?+

The final cell of the bottom row is P(a), exactly what the remainder theorem predicts. That makes synthetic division both a division method and the fastest way to evaluate a polynomial at a point.

Is there a limit on how many sets I can generate?+

No. Problems are produced on demand, free, with no registration. Reload for a fresh set whenever you want one.

⌐Synthetic Division Practice Problems