Remainder Theorem Calculator

Evaluate P(a) instantly, obtain quotient and remainder when dividing a polynomial by (x–a). Perfect for algebra, precalculus, and beyond.

Remainder Theorem: For a polynomial P(x), the remainder when divided by (x – a) is simply P(a).

If P(a) = 0, then (x – a) is a factor of P(x).

Use ^ for exponents, * for multiplication. Variables: x only. Examples: x^3-2x+1, 3x^4+5x^2-2
x³–6x²+11x–6
x²–5x+6
2x³–x²–7x+3
x⁴–16
3x²+2x–1
x⁵–1
The number a in the divisor (x – a).
X‑axis range around the point a.
Computing polynomial division...

Understanding the Remainder Theorem

The Remainder Theorem is a fundamental result in algebra that connects polynomial evaluation with polynomial division. It states:

If a polynomial P(x) is divided by (x – a), the remainder is P(a).

In other words, there exists a polynomial Q(x) (the quotient) such that:

P(x) = (x – a)·Q(x) + R, where R = P(a).

Proof Outline

Consider the division algorithm: for any polynomial P(x) and linear divisor (x – a), there exist unique Q(x) and constant R such that P(x) = (x – a) Q(x) + R. Substituting x = a gives P(a) = (a – a) Q(a) + R = R. ∎

The Factor Theorem – A Direct Corollary

If P(a) = 0, then the remainder is zero, so (x – a) divides P(x) exactly. This is known as the Factor Theorem:

(x – a) is a factor of P(x) ⇔ P(a) = 0.

This is the basis for finding roots and factoring polynomials.

Worked Examples

Example 1: Let P(x) = x³ – 6x² + 11x – 6 and a = 2. Then P(2) = 8 – 24 + 22 – 6 = 0, so (x – 2) is a factor. Division yields Q(x) = x² – 4x + 3, which factors further as (x – 1)(x – 3).

Example 2: P(x) = 2x³ + 3x² – 5x + 7, a = –1. P(–1) = –2 + 3 + 5 + 7 = 13, so remainder = 13. The division statement is P(x) = (x + 1)·(2x² + x – 6) + 13.

Synthetic Division – A Fast Algorithm

Synthetic division is a shorthand method for dividing a polynomial by (x – a). It uses only the coefficients and is especially efficient when checking potential roots. For the polynomial 2x³ + 3x² – 5x + 7 and a = –1:

Step Coefficients Operation
Bring down 2, 3, –5, 7 2
Multiply by –1 –2
Add to next 3 + (–2) = 1
Multiply by –1 –1
Add to next –5 + (–1) = –6
Multiply by –1 6
Add to last 7 + 6 = 13 (remainder)

The quotient coefficients are 2, 1, –6 → Q(x) = 2x² + x – 6.

Historical Context

The Remainder Theorem is a special case of the polynomial remainder theorem, which was known to Chinese mathematicians in the 13th century (Zhu Shijie) and later formalized by Étienne Bézout in the 18th century. It plays a crucial role in the development of algebra and numerical analysis.

Why the Remainder Theorem Matters

  • Root Finding: Test integer candidates quickly using synthetic division.
  • Factoring: Once a root is found, factor the polynomial and reduce its degree.
  • Graph Interpretation: The point (a, P(a)) lies on the graph; if P(a)=0, it's an x‑intercept.
  • Partial Fractions: Remainders help decompose rational functions.
  • Error Analysis: In numerical methods, the remainder bounds the approximation error.

Common Pitfalls

  • Sign Errors: In the divisor (x – a), note that a can be negative. For (x + 3), a = –3.
  • Missing Terms: Ensure polynomials include zero coefficients for missing degrees (e.g., x³ – 1 should be treated as x³ + 0x² + 0x – 1).
  • Non‑integer a: The theorem holds for any real (or complex) a, not just integers.

Extensions

The theorem generalizes to division by higher‑degree polynomials: the remainder when dividing by a quadratic (x² + bx + c) will be of degree less than 2, and can be found by evaluating the polynomial at the roots (if they are real). This leads to the Remainder Theorem for Quadratic Divisors and beyond.

Calculator implementation notes: This tool uses the symbolic algebra engine nerdamer.js to perform exact polynomial division and evaluation. It handles rational coefficients, large exponents, and provides both the quotient and remainder in symbolic form. The graph helps visualize the relationship between the polynomial and the point (a, P(a)).

Frequently Asked Questions

Yes, the remainder theorem holds for any real (or complex) number a. The division algorithm works for any constant a.

The remainder theorem specifically applies to linear divisors. For other divisors, you need full polynomial division, though similar ideas (e.g., evaluating at roots) can be used for quadratic divisors.

It uses the symbolic algebra engine nerdamer. The built‑in divide method returns the quotient and remainder exactly.

For simplicity, this tool expects the variable to be x. Other letters might not be recognized.

The theorem can be adapted: dividing by (ax – b) gives remainder P(b/a). You can first factor out a: (ax – b) = a(x – b/a). The remainder after dividing by (x – b/a) is P(b/a), and then you must account for the factor a in the quotient.