Remainder theorem
Find the remainder when a polynomial is divided by (x−a).
This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.
Core idea
Overview
The Remainder Theorem provides a direct method for calculating the remainder of a polynomial division by a linear factor without performing the full division process. It states that when a polynomial f(x) is divided by a linear divisor in the form (x - a), the remainder is equal to the value of the polynomial evaluated at a.
When to use: Use this theorem when you need to find the remainder of a polynomial division specifically by a linear divisor of the form (x - a). It is a highly efficient shortcut when the quotient itself is not required, saving time compared to long or synthetic division.
Why it matters: This theorem forms the logical basis for the Factor Theorem, which is essential for solving algebraic equations and factoring high-degree polynomials. In computational science, it helps in the development of algorithms for error detection and polynomial interpolation.
Symbols
Variables
a = Value a, f(a) = Function Value
Walkthrough
Derivation
Derivation of the Remainder Theorem
The remainder theorem states that dividing P(x) by (x-a) leaves remainder P(a).
- P(x) is a polynomial.
- Division is by a linear factor (x-a).
Write the Division Identity:
Polynomial division gives quotient Q(x) and constant remainder R.
Substitute x=a:
Evaluate the identity at x=a.
Simplify to Get the Remainder:
The \((a-a)\) term is zero, so the quotient vanishes and only the remainder remains.
Result
Source: OCR A-Level Mathematics — Pure (Algebra)
Free formulas
Rearrangements
Solve for
Remainder theorem
To make f(a) the subject of the Remainder theorem, identify that f(a) is already isolated on the right-hand side and swap the sides of the equation to position the subject on the left.
Difficulty: 2/5
The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.
Visual intuition
Graph
The graph of the remainder theorem plots the remainder $R$ on the y-axis against the constant $a$ from the divisor $(x-a)$, resulting in a linear relationship defined by the polynomial function $f(a) = R$. The line passes through the point $(a, f(a))$, demonstrating that the remainder is directly equivalent to the value of the polynomial at $a$. This linear representation illustrates the Remainder Theorem's core principle that evaluating a polynomial at a specific value determines the remainder upon division by $(x-a)$.
Graph type: linear
Why it behaves this way
Intuition
Imagine the polynomial's graph: the remainder when divided by (x-a) is simply the y-value of the graph at the point where x=a.
Signs and relationships
- a (in f(a)): The value 'a' is derived from the linear divisor (x-a). It represents the specific value of x that makes the divisor equal to zero. This connection is crucial: if the divisor were (x+a), the value to substitute into f(x)
Free study cues
Insight
Canonical usage
The Remainder Theorem is used to find a numerical value, which is the remainder of a polynomial division. This value is inherently dimensionless in a mathematical context.
Common confusion
Students might incorrectly attempt to assign physical units to the remainder itself, even when the polynomial's variables represent physical quantities.
Dimension note
The remainder obtained from the Remainder Theorem is a numerical value. While the variables within a polynomial might represent physical quantities with units in applied contexts, the theorem itself operates on the
One free problem
Practice Problem
Calculate the remainder when the polynomial f(x) = x³ - 4x² + 2x - 5 is divided by (x - 3).
Solve for: fa
Hint: Substitute the value x = 3 into the function f(x).
The full worked solution stays in the interactive walkthrough.
Where it shows up
Real-World Context
CRC checks in computing (similar concept).
Study smarter
Tips
- Always identify 'a' by looking at the divisor; for (x - 5), a = 5, but for (x + 5), a = -5.
- Ensure the divisor is linear; if it is quadratic or higher, this specific theorem cannot be used directly.
- Be careful with negative bases when raising 'a' to even or odd powers during substitution.
Avoid these traps
Common Mistakes
- Evaluating f(-a) instead of f(a).
- Confusing divisor (x-a).
Common questions
Frequently Asked Questions
The remainder theorem states that dividing P(x) by (x-a) leaves remainder P(a).
Use this theorem when you need to find the remainder of a polynomial division specifically by a linear divisor of the form (x - a). It is a highly efficient shortcut when the quotient itself is not required, saving time compared to long or synthetic division.
This theorem forms the logical basis for the Factor Theorem, which is essential for solving algebraic equations and factoring high-degree polynomials. In computational science, it helps in the development of algorithms for error detection and polynomial interpolation.
Evaluating f(-a) instead of f(a). Confusing divisor (x-a).
CRC checks in computing (similar concept).
Always identify 'a' by looking at the divisor; for (x - 5), a = 5, but for (x + 5), a = -5. Ensure the divisor is linear; if it is quadratic or higher, this specific theorem cannot be used directly. Be careful with negative bases when raising 'a' to even or odd powers during substitution.
References
Sources
- Wikipedia: Remainder theorem
- Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
- Britannica: Remainder theorem
- Stewart, James. Calculus: Early Transcendentals
- Stewart, Redlin, and Watson, Precalculus: Mathematics for Calculus, 7th ed.
- OCR A-Level Mathematics — Pure (Algebra)