Remainder theorem
Find a linear-division remainder without performing long division.
Remainder theorem is one of 3 polynomials formulas in the algebra section of this library, and it is used at high school level.
Why remainder theorem works
Division writes the polynomial as the divisor times a quotient plus a remainder, and dividing by something of degree 1 forces the remainder to be a plain constant. Substituting the value that makes the divisor zero wipes out the quotient term entirely, leaving the function value equal to the remainder.
What each symbol means
$f$ is a polynomial and $a$ is the zero of the divisor.
Remainder theorem: when it holds
The divisor must have the form $x-a$; $f(a)=0$ exactly when $x-a$ is a factor.
When it stops applying
It only reports a constant remainder, which requires a divisor of degree 1. Dividing by a quadratic such as x squared + 1 can leave a remainder that still contains x, and no single substitution can produce that expression.
Remainder theorem: a worked example
For $f(x)=x^2+1$ divided by $x-2$, the remainder is $f(2)=5$.
The mistake to avoid
What people do: Substituting the number that appears in the divisor, using -3 for a divisor of x - 3.
Why it goes wrong: The theorem uses the value that makes the divisor zero, which is 3 here. For x cubed - 4x + 1, the correct remainder is 16, while substituting -3 gives -14.
Do this instead: Solve the tiny equation x - a = 0 first and substitute that root, so x + 5 sends you to -5 and x - 3 sends you to 3.
Remainder theorem: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $f$ is a polynomial and $a$ is the zero of the divisor.
- Check the conditions before substituting. The divisor must have the form $x-a$; $f(a)=0$ exactly when $x-a$ is a factor.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Polynomials
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where remainder theorem comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Factoring Polynomials — the lesson behind this formula: reverse multiplication to expose roots and structure.
- Zeros Of A Polynomial Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about remainder theorem
How is this different from the factor theorem?
The factor theorem is this theorem in the special case where the remainder is zero. A zero remainder means the divisor divides evenly, so x - a is a factor of the polynomial.
What value do I substitute for a divisor of x + 5?
Use -5, because that is what makes x + 5 zero. For x squared + 3x - 10 the result is 0, so x + 5 divides that polynomial exactly.
Is synthetic division doing the same thing?
It is the same arithmetic organised into a table, and its final entry is this remainder. Synthetic division also hands you the quotient, which a single substitution does not.
Can I use it to hunt for roots?
Yes, that is its main use. Test candidate values one by one, and any value that returns zero is a root, which lets you factor the polynomial down to something smaller.