Remainder and Factor Theorems
p(c) is the remainder when dividing by x−c; zero remainder means a factor.
Key idea
p(c) is the remainder when dividing by x−c; zero remainder means a factor.
In depth
Dividing P(x) by (x − c) leaves remainder P(c). So (x − c) is a factor exactly when P(c) = 0. It connects roots, factors and evaluating.
Worked example
P(x) = x³ − 2x + 1: P(1) = 0, so (x − 1) is a factor.
Formula
Real-world uses
- Checking if a polynomial model has a particular root, such as a break-even point
Common mistake
Evaluating P(−c) instead of P(c).
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 Remainder and Factor Theorems?
p(c) is the remainder when dividing by x−c; zero remainder means a factor. It is a topic at the advanced level in Algebra.
What is the formula for Remainder and Factor Theorems?
The key formula is p(c) = 0 ⇔ (x−c) | p.
Where is Remainder and Factor Theorems used in real life?
Real-world examples: Checking if a polynomial model has a particular root, such as a break-even point.
What should I learn before Remainder and Factor Theorems?
Make sure you are comfortable with Synthetic Division, the previous topic in Algebra.
What is a common mistake with Remainder and Factor Theorems?
Evaluating P(−c) instead of P(c).
How can I practise Remainder and Factor Theorems?
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 Remainder and Factor Theorems?
Move on to Rational Root Theorem 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.