Arithmetic
Additive Identity
Arithmetic Progression
Associative Property
Averages
Brackets
Closure Property
Commutative Property
Conversion of Measurement Units
Cube Root
Decimal
Distributivity of Multiplication over Addition
Divisibility Principles
Equality
Exponents
Factors
Fractions
Fundamental Operations
H.C.F / G.C.D
Integers
L.C.M
Multiples
Multiplicative Identity
Multiplicative Inverse
Numbers
Percentages
Profit and Loss
Ratio and Proportion
Simple Interest
Square Root
Unitary Method
Algebra
Cartesian System
Order Relation
Polynomials
Probability
Standard Identities & their applications
Transpose
Geometry
Basic Geometrical Terms
Circle
Curves
Angles
Define Line, Line Segment and Rays
Non-Collinear Points
Parallelogram
Rectangle
Rhombus
Square
Three dimensional object
Trapezium
Triangle
Quadrilateral
Trigonometry
Trigonometry Ratios
Data-Handling
Arithmetic Mean
Frequency Distribution Table
Graphs
Median
Mode
Range

Videos
Solved Problems
Home >> Polynomials >> Remainder Theorem in Polynomial >>

Remainder Theorem in Polynomial

Algebraic Expression Algebraic Equation Ordering of Polynomials Types of Polynomials Addition of Polynomials
Subtraction of Polynomials Multiplication of Polynomials Division of Polynomials Types of Degree / Powers in Polynomials Difference between Polynomials of Integers & Rationals
Find Value of Polynomial Find Zero of Polynomial Remainder Theorem in Polynomial Linear Equations Quadratic Equation
Factoring of Quadratic Polynomials



Before you understand Remainder Theorem, you are advised to read:

What is Dividend, Divisor, Quotient & Remainder ?
What is Degree of a Polynomial ?
How to Find Value Polynomial ?
How to divide Polynomial by Binomial ?

(This Remainder Theorem applies to situation where a polynomial is divided by a linear polynomial)

Remainder Theorem says that, " If p(x) is divided by linear polynomial (x - a), then remainder will always be p(a) ".
Here, p(x) can be any polynomial of degree greater than or equal to zero. And 'a' can be any real number.

Lets prove this Theorem with the help of following examples:

Example 1: Divide (x3 + 4) by (x - 1)
Solution: As per the given question:
Polynomial = Dividend= (x3 + 4)
Linear Polynomial = Divisor = (x - 1)

Division is done as shown below:



Hence, when (x3 + 4) is divided by (x - 1), we get:
Quotient = (x2 + x + 1)
Remainder = 5.... (Statement 1)

Now, let's apply remainder theorem on the same example:

Remainder Theorem says that, " If p(x) is divided by linear polynomial (x - a), then remainder will always be p(a) ".

Apply the values from given example and we get:
Polynomial p(x) = (x3 + 4)
Linear Polynomial (x - a) = (x - 1)
So value of a = 1
And as per remainder theorem, x = a = 1

Now find the value of p(x) = (x3 + 4), where x = 1 and we get:
p(1) = (1)3 + 4
= 1 + 4 = 5 .... (Statement 2)

Hence, as per statement 1 and 2, Remainder is same i.e. 5.

This proves the remainder theorem.

Copyright@2022 Algebraden.com (Math, Algebra & Geometry tutorials for school and home education)