Remainder Theorem
The Remainder Theorem is a simple yet powerful tool in algebra that helps you quickly find the remainder when dividing a polynomial by a linear polynomial, such as (x - a). Instead of performing long or synthetic division, you can use this theorem to substitute the polynomial and get the remainder directly. It simplifies many complex calculations and is especially useful in solving polynomial equations and identifying factors.
Now let's learn about the Reminder theorem, its proof, and others in detail in this article.
Table of Content
What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial f(x) is divided by a linear polynomial of the form (x – a), the remainder is simply the value of the function evaluated at x = a. This simplifies polynomial division, allowing us to bypass long division in certain cases.
For example, if you have a polynomial P(x) and divide it by x – a, the remainder will be P(a). This is crucial in determining whether a polynomial is divisible by another without performing complex operations. The key formula for the Remainder Theorem is:
Remainder = f(a) when dividing f(x) by (x – a).
This theorem helps in simplifying polynomial calculations and is often used in conjunction with the Factor Theorem.
Proof of the Remainder Theorem
To prove the Remainder Theorem, we start with the polynomial division identity:
f(x) = (x – a) * q(x) + r
Where:
- f(x) is the dividend
- x – a is the divisor
- q(x) is the quotient
- r is the remainder
Substituting x = a into the equation gives:
f(a) = (a – a) * q(a) + r
Since a – a = 0, this simplifies to f(a) = r. Therefore, the remainder is the value of the polynomial evaluated at x = a. This provides a straightforward and efficient way to calculate remainders in polynomial division, especially when dealing with linear divisors.
Remainder Theorem Formula
For any polynomial function f (x) when divided by the linear polynomial (x-a) then the remainder is always equal to f (a).
f (x) = (x-a) × q (x) + r (x)
Where r(x) is the remainder of the polynomial.
The image given below express the remainder theorem.
Remainder Theorem of Polynomial
How to use the Remainder theorem? can easily be explained with the help of the example given below.
Example: Divide 2x3 + 3x2 + 4x + 5 by x + 2
Solution:
Given,
Dividend = p(x) = 2x3 + 3x2 + 4x + 5
Divisor = s(x) = (x + 2)
Here,
Quotient = q(x) =2x2 - x + 6
Remainder = r(x) = -7
Verification:
Given,
Divisor = (x + 2)
Let x + 2 = 0
x = -2
According to Remainder Theorem, substituting x = -2 in p(x),
Now, Remainder = p(-2)
Thus, Remainder Theorem is verified.
Alternate Method
We know that any polynomial p(x) can easily be written as,
p(x) = (x - a)·q(x) + r
Taking x = a in the above equation we get only the remainder,
similarly, for any p(x)
p(x) = (x + 2).q(x) + r
if we put x = -2 we get the remainder of the above equation.
Thus, p(-2) is the remainder.
Hence proved.
Applications of the Remainder Theorem
The Remainder Theorem finds applications in various fields, including mathematics, computer science, and engineering. Some key applications include:
- Simplifying polynomial division: The theorem eliminates the need for long division in certain cases.
- Factoring polynomials: Using the theorem with the Factor Theorem allows for easier factorization.
- Solving polynomial equations: It helps in checking whether a given number is a root of the polynomial.
- Signal processing: In engineering, polynomials are used to model signals, and the Remainder Theorem aids in simplifying these models.
Remainder Theorem and Factor Theorem
The Factor Theorem is a direct extension of the Remainder Theorem. It states that if the remainder is 0, then (x – a) is a factor of the polynomial. In other words, if f(a) = 0, the polynomial f(x) is divisible by (x – a).
For instance, in the previous example, since P(1) = 0, we know that x – 1 is a factor of the polynomial P(x).
The Factor Theorem is commonly used in algebra to identify factors of polynomials, enabling further simplifications, such as breaking down higher-degree polynomials into simpler factors.
Remainder Theorem and Factor Theorem are the basic theorems in mathematics and the basic difference between them is stated in the table given below:
Difference Between Remainder Theorem and Factor Theorem | ||
|---|---|---|
Feature | Remainder Theorem | Factor Theorem |
Definition | According to, Remainder Theorem for any polynomial p(x) when divided by x - a the remainder is p(a). | According to Factor Theorem if (x - a) is a factor of p(x) then this is true only if f(a) = 0. |
Purpose | This theorem will find the remainder when a polynomial is divided by a linear expression. | This theorem determines whether a given expression is a factor of a polynomial or not. |
Use-case | It is primarily used to find the remainder when dividing polynomials. | It is used to find factors of polynomials, which can be useful for polynomial factorization and solving polynomial equations. |
Relation to Roots | It does not directly provide any information about the roots of a polynomial. | It helps to find roots of a polynomial. |
Application | Remainder Theorem helps us to find the remainder of the polynomial without actually dividing it. | Factor theorem helps us to find the factor of the given polynomial. |
Articles related to Remainder Theorem:
Remainder Theorem Examples
Example 1: Find the remainder when p(x) = x4 - x3 + x2 - 2x + 1 is divided by g(x) = x - 2.
Solution:
Given,
p(x) = x4 - x3 + x2 - 2x + 1
g(x) = x-2
Using Remainder theorem,
p(2) = (2)4 - (2)3 + (2)2 - 2(2) + 1
= 9
Thus, the remainder when p(x) is divided by g(x) then we get remainder as, 9
Example 2: Find the root of the polynomial x2 - 5x + 4
Solution:
We know that if for any p(x) we get p(a) = 0, then x-a is the factor of p(x) or a is the root of equation.
Given,
f(x) = x2 - 5x + 4
By hit and try method.
f(4) = 42 - 5(4) + 4
f(4) = 20 - 20
= 0
Since f(4) = 0, x = 4 is a root of the equation, and (x − 4) is a factor of x2 − 5x + 4.
Example 3: Find the remainder when t3 - 2t2 + 4t + 5 is divided by t - 1.
Solution:
Given,
p(t) = t3 - 2t2 + 4t + 5
g(t) = t - 1
Using Remainder theorem,
g(1) = (1)3 - 2(1)2 + 4 + 5
= 8
By the Remainder Theorem, 8 is the remainder when t3 - 2t2 + 4t + 5 is divided by t - 1
Example 4: Find the remainder when x3 - x2 + 2 is divided by x - 2.
Solution:
Given,
p(x) = x3 - x2 + 2
g(x) = x-2 = 0
x = 2
Using Remainder theorem,
g(2) = (2)3 - (2)2 + 2
= 6
By the Remainder Theorem, 6 is the remainder when x3 - x2 + 2 is divided by x - 2
Example 5: Find the root of the polynomial 3x2 - 7x + 2
Solution:
We know that if for any p(x) we get p(a) = 0, then x-a is the factor of p(x) or a is the root of equation.
Given,
f(x) = 3x2 - 7x + 2
By hit and try method.
f(2) = 3(2)2 - 7(2) + 2
f(2) = 12 -14 + 2
= 0
Since f(2) = 0, x = 2 is a root of the equation, and (x − 2) is a factor 3x2 - 7x + 2.
Euler Remainder Theorem
For any two co-prime positive integers n and X, Euler's theorem states that,
Xφ(n) ≡ 1 (mod n)
where φ(n) is called Euler's function and its value is given as,
φ(n) = n (1-1/a)×(1-1/b)(1-1/c)
where
- n is a natural number
- n = ap × bq × cr
- a, b, c are prime factors of n
- p, q, and r are positive integers.
Example: Find the Euler function of 21.
Solution:
Factors of 21 are,
21 = 7×3
φ(21) = 21 (1 - 1/7)(1 - 1/3)
= 21 × 6/7 × 2/3
= 12
Thus, the Euler function of 21 is 12
Summary
The Remainder Theorem in algebra establishes a direct connection between polynomial division and polynomial evaluation. It states that when a polynomial P(x) is divided by a linear polynomial x - a, the remainder is equivalent to P(a). This theorem simplifies polynomial evaluation by allowing one to find the remainder of a division by directly substituting a value into the polynomial. Consequently, it is frequently employed to determine roots of polynomials and to factorize them efficiently. Together with the Factor Theorem and synthetic division, the Remainder Theorem plays a crucial role in polynomial manipulation and problem-solving across various mathematical contexts.
