Polynomials are one of the most important chapters in the world of maths. When kids are interested in solving different questions, they have to learn multiple formulas. One concept is the remainder theorem which students have to learn in order to solve mathematical problems. Therefore, it’s important that its concept would be clear to the students to understand and solve questions easily.
Here is a guide about the remainder theorem, its definition,  proofs, and examples. Check it out:

Remainder Theorem

Finding the reminder with the remainder theorem formula: (px) would be any polynomial of a degree greater than, or equal to one, and can be any real number. If the polynomial (px) is divided by (x – k), then the remainder obtained is (pc). Now that you know what the remainder theorem is, let’s learn about it a bit more.

Remainder Theorem Proof

When we divide (px) simply by the polynomial (x – k ), we get:

As (x – k) has the degree 1,  r (x),  is a reminder as well, and it must have a degree of 0.
Therefore, the remainder is just constant r.  Accordingly, we’ll  get:

If we substitute x  with k, we’ll get: 

Why The Remainder Is constant in the Remainder Theorem?

The remainder theorem proves states that when the given polynomial gets divided by (x – k) the remainder obtained will either be 0 or a degree less than (x – k). As (x – k) is equal to degree 1, the degree of the remainder must be 0. This implies that the remainder is constant.

Therefore, in either case  p(x) = (x – k) q(x) + r(x).  In this case, the remainder r is a real number, probably 0.  Substituting the variable “x” with ‘k’, we get p(k)  =  r as required.  You can visit Cuemath to learn about this topic in detail.

The Steps to divide the polynomial by the non-zero polynomial are:

1. First of all, students need to arrange the polynomials in the decreasing order of degree.
2. Then they need to divide the first term of the dividend by the first term of the divisor to produce the first term of the quotient.
3. Then one needs to multiply the divisor by the first term of the quotient and subtract this product from the dividend to get the reminder.
4. The reminder is the dividend now, and the divisor will remain the same.
5. The students would need to repeat this until the degree of the new dividend is less than the degree of divisor.

Remainder Theorem Example

What will be the remainder when the given polynomial:

Gets divided by x – 1?  The solution is:

Let the dividend be

Let divisor be x –  1

If we equate the divisor to 0, we get:  x  –  1 = 0 and  x  =  1.
Therefore, substituting the value of x in the polynomial –
We get: