Remainder theorem
Remainder Theorem
The Remainder Theorem is a fundamental result in algebra that provides a quick way to calculate the remainder of a polynomial when divided by a linear divisor. It is particularly useful when dealing with polynomial division, simplifying the process significantly.
Understanding the Remainder Theorem
The theorem states that if a polynomial $f(x)$ is divided by a linear divisor of the form $(x - c)$, then the remainder of this division is simply $f(c)$. In other words, if you substitute the value $c$ into the polynomial $f(x)$, the result is the remainder when $f(x)$ is divided by $(x - c)$.
Mathematical Formulation
If $f(x)$ is a polynomial and we divide it by $(x - c)$, then:
$$ f(x) = (x - c) \cdot q(x) + r $$
where:
- $q(x)$ is the quotient polynomial.
- $r$ is the remainder, which is a constant since $(x - c)$ is a linear polynomial.
According to the Remainder Theorem, $r = f(c)$.
Example of the Remainder Theorem
Let's consider the polynomial $f(x) = 2x^3 - 3x^2 + 4x - 5$ and we want to find the remainder when it is divided by $(x - 2)$.
Using the Remainder Theorem, we simply evaluate $f(2)$:
$$ f(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 $$ $$ f(2) = 2(8) - 3(4) + 8 - 5 $$ $$ f(2) = 16 - 12 + 8 - 5 $$ $$ f(2) = 7 $$
So, the remainder when $2x^3 - 3x^2 + 4x - 5$ is divided by $(x - 2)$ is 7.
Table of Differences and Important Points
| Feature | Polynomial Division | Remainder Theorem |
|---|---|---|
| Purpose | To divide one polynomial by another, obtaining a quotient and a remainder. | To quickly find the remainder when a polynomial is divided by a linear factor. |
| Process | Long division or synthetic division must be used to find the quotient and remainder. | Simply substitute the value of $c$ into the polynomial $f(x)$. |
| Result | Provides both quotient and remainder. | Provides only the remainder. |
| Efficiency | Can be time-consuming for complex polynomials. | Very efficient for finding remainders. |
| Applicability | Can be used for dividing by any polynomial. | Only applicable for linear divisors of the form $(x - c)$. |
Using the Remainder Theorem with Higher Degree Polynomials
The Remainder Theorem is not limited to cubic polynomials like in the example above. It can be applied to any polynomial of degree $n$, where $n$ is a positive integer.
Example with a Higher Degree Polynomial
Consider the polynomial $f(x) = x^4 - 2x^3 + x - 3$ and we want to find the remainder when it is divided by $(x + 1)$.
Using the Remainder Theorem, we evaluate $f(-1)$:
$$ f(-1) = (-1)^4 - 2(-1)^3 + (-1) - 3 $$ $$ f(-1) = 1 + 2 - 1 - 3 $$ $$ f(-1) = -1 $$
So, the remainder when $x^4 - 2x^3 + x - 3$ is divided by $(x + 1)$ is -1.
Conclusion
The Remainder Theorem is a powerful tool in algebra that simplifies the process of finding remainders in polynomial division. It is particularly useful for linear divisors and can save a significant amount of time compared to performing long division. Understanding and applying the Remainder Theorem can be very beneficial, especially in examinations where time efficiency is crucial.