Remainder Theorem
In short: The remainder theorem is the shortcut that turns a long division by a linear binomial into one substitution: the number left over is simply the polynomial evaluated at the zero of the divisor.
Algebra 2AlgebraA-APR.B.2
What is Remainder Theorem?
The remainder theorem answers a practical question: what is left over when a polynomial is split by a linear expression? You do not need to carry out the whole division to find out.
Write the divisor as \(x – c\). The leftover is the constant \(f(c)\), so that \(f(x) = (x – c)\,q(x) + f(c)\) with \(q(x)\) the quotient. A divisor like \(x + 4\) becomes \(x – (-4)\), which means you evaluate at \(c = -4\).
How to spot it
- A problem asks for the remainder after dividing by a linear factor such as x – 3 or x + 5.
- You are told to find f(c) for a polynomial and to explain what that value means for division.
- Long division would be slow, yet only the leftover number is needed, not the whole quotient.
- A question asks whether a given binomial divides a polynomial evenly, which means a remainder of zero.
The method
- Write the divisor in the form \(x – c\) and read off \(c\); for \(x + 4\) take \(c = -4\).
- Substitute \(c\) into the polynomial in place of every \(x\).
- Evaluate with care, keeping track of negative signs and exponents: the result is \(f(c)\).
- State the answer: by the remainder theorem the leftover is \(f(c)\). If it is \(0\), the divisor is a factor.
Remainder Theorem: solved example
Problem
Find the remainder when \(f(x) = 2x^3 – 5x^2 + 4x – 9\) is divided by \(x – 3\).
Solution
The divisor is \(x – 3\), so \(c = 3\). Substitute: \(f(3) = 2(27) – 5(9) + 4(3) – 9\).
Compute each piece: \(54 – 45 + 12 – 9 = 12\). The remainder is 12, and no long division was needed.
Try it
Question 1
Find the remainder when \(p(x) = x^4 – 3x^2 + 7\) is divided by \(x + 1\).
Show the answer
Here \(c = -1\). \(p(-1) = 1 – 3 + 7 = 5\), so the remainder is 5.
Question 2
Does \(x – 2\) divide \(h(x) = x^3 – 4x^2 + x + 6\) evenly?
Show the answer
\(h(2) = 8 – 16 + 2 + 6 = 0\). The remainder is 0, so yes, \(x – 2\) divides it evenly.
Remainder Theorem vs. Factor Theorem
The remainder rule gives the leftover for any value of \(c\): it always answers “what remains?” The factor rule is the special case in which that leftover is \(0\). If \(f(c) = 0\), then \(x – c\) is a factor; if \(f(c) \neq 0\), the number \(f(c)\) is still the remainder, but the binomial is not a factor.
Keep going
Lessons
Worksheets and quizzes
- Dividing Polynomials and the Remainder Theorem – Worksheet
- Dividing Polynomials and the Remainder Theorem – Quiz
- Polynomial Division and the Factor Theorem – Worksheet
- Polynomial Division and the Factor Theorem – Quiz
Related words
- Factor Theorem
- Polynomial Long Division
- Synthetic Division
- Zeros of a Function
- Rational Root Theorem
Whole dictionary: A–Z · Algebra 2 · Algebra