Remainder and Factor Theorems
Evaluating p(a) gives the remainder when dividing by (x − a).
About this topic
The Remainder Theorem states that when a polynomial f(x) is divided by x − c, the remainder is f(c). So you can evaluate the polynomial instead of dividing. The Factor Theorem follows: x − c is a factor of f(x) exactly when f(c) = 0.
These theorems link roots and factors. For f(x) = x³ − 4x² + x + 6, evaluating f(2) gives 8 − 16 + 2 + 6 = 0, so x − 2 is a factor, and dividing leaves x² − 2x − 3, which factors further into (x − 3)(x + 1). The roots are 2, 3 and −1. If instead f(1) = 4, the remainder when dividing by x − 1 is 4.
These ideas make it easy to test candidate roots and to build a polynomial from its zeros, such as writing a cubic that crosses the x-axis at specific points. Synthetic division is a fast shortcut that gives both the quotient and f(c).
Mind the sign: dividing by x + 3 means c = −3, so you evaluate f(−3). A remainder of zero does not mean the polynomial is zero; it means the divisor is a factor. Also remember that a polynomial of degree n has at most n real roots.
Key ideas
- If p(a) = 0 then (x − a) is a factor
- Use synthetic division for speed
- Find roots by testing candidates
Quick check
Use cases: where Remainder and Factor Theorems helps
- Homework: review the key idea behind Remainder and Factor Theorems before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Remainder and Factor Theorems without notes.
- Visual learning: play with the interactive lab to see how Remainder and Factor Theorems changes when you adjust its inputs.
- Teaching a friend: explain Remainder and Factor Theorems in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Remainder and Factor Theorems?
Evaluating p(a) gives the remainder when dividing by (x − a).
What will I learn in Remainder and Factor Theorems?
If p(a) = 0 then (x − a) is a factor. Use synthetic division for speed. Find roots by testing candidates.
Is Remainder and Factor Theorems part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Remainder and Factor Theorems lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Remainder and Factor Theorems?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
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