The Basic Concept of Remainder Theorem

When someone questions what is remainder theorem, let us inform that this particular remainder theorem is used to evaluate the remainder whenever a polynomial is divided by a linear polynomial.  

In case you are wondering what is remainder, then let us understand with the help of an example-

For instance, a particular number of items are being divided into multiple groups. Moreover, each group will have an equal number of items, the number of items left in the last without any group is known as remainder.

Let’s learn the basic concept of the remainder theorem.

What Is Meant By The Remainder Theorem?

Here is the definition of what is remainder theorem – the theory that helps in solving the division of a polynomial by another linear polynomial. There is no need for carrying out all steps required to complete the division algorithm. 

The remainder theorem can be stated as follows:

Let’s say that a polynomial p(x) is being divided by a linear polynomial x-a, the quotient in such a case is given by q(x) and a polynomial remainder r(x). 

Here, 

p(x) = p is a polynomial whose variable is x

x-a = a is a rational number 

q(x) = q is quotient

r(x) = r is remainder

What Is The Formula For The Remainder Theorem?

The general remainder theorem formula can be mathematically expressed as:

p(x) / x-a = q(x) + r(x)

The remainder theorem formula: 

When p(x) is divided by a polynomial x-a, then the remainder can be expressed as p(a). Or, when p(x) is divided by (ax + b), then the remainder can be expressed as P(- b / a).

Stating The Remainder Theorem Proof:

The main element on which the remainder theorem is dependent is that a polynomial is comprehensively divisible, at least once by its factor so that a smaller value of the polynomial can be obtained and “a” remainder of zero. This is the most simple way by which an individual can know whether the value of “a” is a root of polynomial p(x).

This is when p(x) is divided by (x-a) and thus, an equation is obtained:

p(x) = (x-a).q(x) + r(x)

It is a universal fact that Dividend = (Divisor x Quotient) + Remainder

In case the r(x) as mentioned in the equation is a constant “r”, then the following equation is obtained,

p(x) = (x-a).q(x) + r

Now observe what happens if ‘x’ is taken equal to the ‘a’:

p(a) = (a-a).q(a) + r

p(a) = (0).q(a) + r

p(a) = r

Hence, proved.

What Is The Procedure Of Dividing A Polynomial By Another Non-Zero Polynomial?

Here mentioned are the steps that are to be followed according to the remainder theorem when dividing a polynomial by another non-zero polynomial:

1. First, you are required to arrange the polynomials, both dividend and divisor in the decreasing order according to the degree.

2. Now the first term of the dividend is to be divided by the first term of the divisor so that the first term of the quotient can be produced.

3. Next, you need to multiply the divisor by the quotient’s first term and subtract the calculated product from the divided so that the remainder can be procured.

4. The obtained remainder next acts as the dividend and the divisor remains the same.

5. Lastly, you are required to repeat the process starting from the first step until the degree of the new dividend is less than the divisor’s degree. 

Important Notes To Remember:

• The general formula of the remainder theorem is: p(x) / x-a = q(x) + r(x)

• The universal formula one can use to check the division is Dividend = (Divisor x Quotient) + Remainder

• Whenever a polynomial, say p(x) is divided by another linear polynomial, say q(x) whose zero is x = k, the remainder will be given by r = p(k)