Question 1

What is synthetic division?

Accepted Answer

Synthetic division is a shorthand algorithm for dividing a polynomial p(x) by a linear binomial (x − k). It avoids writing variable names by working only with the polynomial's coefficients in a tabular format. You write k in a box on the left, list the coefficients of p(x) across the top, then proceed row by row: bring down, multiply by k, add, repeat. The bottom row gives the coefficients of the quotient plus the remainder as the last entry.

Question 2

When should I use synthetic division instead of long division?

Accepted Answer

Use synthetic division whenever the divisor is a linear binomial (x − k) or can be rewritten as one, for example (x + 5) = (x − (−5)). Synthetic division is faster and less error-prone than polynomial long division for these cases. Use polynomial long division when the divisor has degree two or higher, such as (x² + 2x − 1), because synthetic division only works for linear divisors.

Question 3

What is the Remainder Theorem?

Accepted Answer

The Remainder Theorem states that when a polynomial p(x) is divided by (x − k), the remainder equals p(k). You can therefore evaluate p(k) without substituting k into the polynomial — just perform synthetic division and read the last number in the bottom row. For example, to find p(3) for p(x) = x³ − 4x + 1, synthetic division with k = 3 gives remainder 16, so p(3) = 16.

Question 4

What is the Factor Theorem and how does it relate to synthetic division?

Accepted Answer

The Factor Theorem is a consequence of the Remainder Theorem: (x − k) is a factor of p(x) if and only if p(k) = 0, i.e., the remainder is zero. When synthetic division yields a remainder of 0, the divisor (x − k) divides p(x) exactly, and k is a root of p(x). The quotient polynomial from the bottom row can then be factored further, making synthetic division a practical tool for fully factoring higher-degree polynomials.

Question 5

How do I use synthetic division to divide by (x + k)?

Accepted Answer

(x + k) = (x − (−k)), so set the synthetic division value to −k (the negative of the constant term). For example, to divide by (x + 3), use k = −3. Write −3 in the box and proceed normally with the coefficients. This sign flip is the only adjustment needed — the algorithm is identical. Always rewrite the divisor in the form (x − k) and read off k before starting.

Polynomial	Divisor	k value	Quotient	Remainder	Factor?
x²−5x+6	(x−2)	2	x−3	0	Yes
x²−5x+6	(x−3)	3	x−2	0	Yes
x³−8	(x−2)	2	x²+2x+4	0	Yes
x³−6x²+11x−6	(x−1)	1	x²−5x+6	0	Yes
x²+1	(x−1)	1	x+1	2	No
2x³−3x+1	(x+2)	−2	2x²−4x+5	−9	No

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