Question 1

When can I use synthetic division?

Accepted Answer

Only when you are dividing by a linear factor of the form x - c. For divisors like x² + 1 or 2x - 3 you need long division, or you can divide by x - 3/2 and adjust.

Question 2

What number goes in the box?

Accepted Answer

The zero of the divisor. For x - 2 the box gets 2. For x + 5 the box gets -5, because x + 5 = x - (-5). Getting this sign wrong is the most common synthetic division error.

Question 3

Why does the calculator add zeros to my polynomial?

Accepted Answer

Every power of x needs a column, even if the term is missing. Dividing x³ - 7x + 6 uses coefficients 1, 0, -7, 6 because there is no x² term. Skipping the 0 shifts every column and gives the wrong answer.

Question 4

What does the remainder tell me?

Accepted Answer

By the remainder theorem, the remainder equals the polynomial evaluated at c. If it is 0, then x - c is a factor and c is a root. That is why synthetic division is the fast way to test possible rational roots.

Question 5

How do I write the final answer?

Accepted Answer

Quotient plus remainder over the divisor. For 2x³ - 3x + 5 divided by x - 2 the answer is 2x² + 4x + 5 + 15/(x - 2). If the remainder is 0, just write the quotient.