State and Prove Remainder Theorem – Polynomials

15 Sep
Class 9 - Polynomials, Math - Class 9 | NCERT, Theorems, , , , , , ,

Remainder Theorem

Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

Proof

Let p(x) be any polynomial of degree >= 1

Dividing p(x) by x-a we will get

\( \begin{aligned} &p(x) = (x-a)q(x)+ r(x) \\\\ &\text{or we can write}\\\\&p(x) = (x-a)q(x)+ r \ \ …(1)\end{aligned}\)

Where q(x) and r are quotient and remainder respectively. 

Remainder r will be a constant because either r is 0 or deg(r) < deg (x-1).

Not able to get it? Watch the video tutorial or Click here to read Why Remainder is a constant?

Put x=a in equation (1)

\( \begin{aligned} p(a) &= (a-a)q(a)+ r \\ \Rightarrow \ \ p(a) &= 0+ r \\ \Rightarrow \ \ r &= p(a) \end{aligned}\)

Hence proved.

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Why Remainder is a constant?

The deg (x-a) is 1. If we divide any polynomial of deg >= 1 by x – a, there will be two possibilities:

  1. It will be divided completely and the remainder will be 0 
  2. Remainder will be non-zero

If the remainder is 0, then we are happy 💃🕺 with what we want to prove. Obviously, 0 is also a constant.  

Let us now talk 📣 🗣️ about the second scenario – Remainder is non-zero. 

We can keep on dividing a polynomial with x – a until we get a remainder that has degree smaller than degree of x – a. 

Only when deg (r) < deg (x – a), we can’t divide further. 

This means the deg (r) = 0 because deg (x – a) = 1. This means remainder is a constant. 

If this is still not clear then either I am not able to explain it well 😵‍💫 🥀 … or you … 😜

More confusion 🤷‍♂️ 🤷‍♀️ Does this mean the Deg (r) = 0?

As we know, that remainder can be 

(1) Non-Zero Constant

(2) Zero 

(1) Non-Zero Constant

The degree of any non-zero constant is 0. You can understand it like this:

\( \begin{aligned} &3 \text { can be written as } 3x^0 \end{aligned}\)

(2) Zero

The degree of 0 is not defined. You can understand it like this:

\( \begin{aligned} &0 \text { can be written as } 0x^0 = 0x^3=0x^7=0x^{565} \end{aligned}\)

Final Deal 💰

After explaining so much if this is still not clear then you need to either say me some “good words” 🤬 $#%&%#🚫 or you must invite me to a have a cup of coffee ☕ so that I can explain it better 👨‍🏫 / 👩‍🏫. Mind it – I will not go out of Gurgaon 🚗 for coffee. 😎

Prerequisite

This is a very important theorem from polynomials perspective. It helps solve very long divisions quite quickly. To understand its significance, you are recommended to learn about the long division of polynomials. That will give you more clarity why this theorem is so important and helpful.

(Topic 2.4 of chapter 2 of class 9 Mathematics NCERT textbook)

Long division method 

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