Synthetic Division
A shortcut for dividing by (x − c).
Key idea
A shortcut for dividing by (x − c).
In depth
A shortcut for dividing by (x − c) using only coefficients. Bring down the leading coefficient, multiply by c, add, and repeat. The final number is the remainder.
Worked example
Divide x³ − 6x² + 11x − 6 by (x − 1): coefficients 1, −6, 11, −6 → 1, −5, 6, 0. Quotient x² − 5x + 6, remainder 0.
Real-world uses
- Quickly checking roots of a polynomial model in engineering and computing
Common mistake
Using c with the wrong sign: for (x + 2), use c = −2.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Synthetic Division?
A shortcut for dividing by (x − c). It is a topic at the advanced level in Algebra.
Where is Synthetic Division used in real life?
Real-world examples: Quickly checking roots of a polynomial model in engineering and computing.
What should I learn before Synthetic Division?
Make sure you are comfortable with Polynomial Long Division, the previous topic in Algebra.
What is a common mistake with Synthetic Division?
Using c with the wrong sign: for (x + 2), use c = −2.
How can I practise Synthetic Division?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Synthetic Division?
Move on to Remainder and Factor Theorems next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.