# Remainder Theorem – The Most Useful Shortcut in Mathematics!

Everyone enjoys finding a shortcut, whether it is for driving directions or another type of long task. Finding a faster and more efficient way to the same destination makes you feel good because you’ve probably saved time, effort, and/or money. The remainder theorem is one of the more useful of these shortcuts in mathematics.

The Remainder Theorem is an approach to Euclidean polynomial division. According to the remainder theorem, dividing a polynomial P(x) by a factor (x – a) that isn’t an element of the polynomial yields a smaller polynomial and a remainder. This obtained remainder is actually a value of P(x) at x equals a, more specifically P (a). x – a is thus the divisor of P(x) if and only if P(a) = 0. It is used to factor polynomials of any degree in an elegant way.

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The Remainder Theorem starts with a polynomial, say p(x), where “p(x)” is some polynomial p with x as a variable. Then, according to the Remainder theorem, divide that polynomial p(x) by some linear factor x – a, where an is simply some number. We go through a long polynomial division here, and this long division which yields a polynomial q(x) (the variable “q” stands for “the quotient polynomial”) and a polynomial remainder is r. (x). It can be stated as follows:

p(x)/x-a equals q(x) + r(x). If you want to learn this in an interactive and engaging manner, you can join cuemath classes and understand it better.

## Factor Theorem is Opposite to Remainder Theorem

The Factor Theorem is commonly used to factor and find the roots of polynomial equations. This theorem is just the opposite of the remainder theorem. Problems are solved by using synthetic division and then checking for a zero remainder.

When p(x) equals 0, y-x is a polynomial factor. Or, to put it another way, if y-x is a factor of the polynomial, then p(x) = 0.

## How Does the Remainder Theorem Work?

To apply the remainder theorem, we can focus on the following steps:

1. Take note of the given polynomial.

2. Arrange the polynomial in increasing order of power.

We can justify the answer by performing long division or by using the remainder theorem, which is p(x) = (x-c)q(x) + r(x).

## What is the Remainder Theorem Formula?

The general formula for remainder theorem is expressed as p(x) equals (x-c)·q(x) + r(x).

## Procedure for Dividing a Polynomial by a Non-Zero Polynomial

Step 1: The polynomial (dividend and divisor) is arranged in decreasing degree order.

Step 2: In the second step, you need to divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.

Step 3: Now multiply the first term of the quotient by the first term of the divisor and subtract the result from the divided to find the remainder.

Step 4: Using the division, divide the remainder.

Step 5: Repeat Step 4 until you can no longer divide the remainder.

## Real-Life Applications of Remainder Theorem

1. The remainder theorem is known to provide a more efficient method for determining whether certain numbers are polynomial roots.

2. This theorem can improve efficiency when using other polynomial tests, such as the rational roots test.

3. Algebraic knowledge is required for higher math levels such as trigonometry and calculus. Algebra also has a plethora of real-world applications.