Class 9 | Chapter 2 | Polynomials | Factor theorem

Sunil Syal — Wed, 16 Sep 2020 01:07:26 +0000

Factor theorem: If p(x) is a polynomial of degree n > 1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).

Prerequisite

(Topic 2.5 of chapter 2 of class 9 Mathematics NCERT textbook)

Another very important theorem of polynomials. we can also call it an extension of remainder theorem. The understanding of remainder theorem is mandatory to solve this theorem.

Remainder theorem

Chapter 2 – Polynomials

Here you will find complete solution of all questions, concepts, theorems and examples of class 9 chapter 2 (Polynomials) of class 9, NCERT textbook.

There are few very important concepts that students must learn to get a good understanding of polynomials:

All above mentioned tutorials are included in the list displayed below for class 9 chapter 2 (Polynomials). You are strongly advised to watch all these videos thoroughly and do not miss any of these if you really want to get good hold on polynomials.

Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Introduction to Polynomials | Introduction to Algebra | Class 9, Chapter 2, NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 1 | Finding Degree | NCERT Maths

Question 1 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 5 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Class 9 | Chapter 2 | Polynomials Example 2 explained | NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 3 explained | NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 4 explained | NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 5 explained | NCERT Maths

Question 1 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

FACTOR THEOREM Proof Explained | Class 9 | Chapter 2 | Polynomials, 2.5 Factorisation of Polynomials

Example 6 Chapter 2 Class 9 | Polynomials, Factor Theorem Questions | NCERT Maths

Example 7 Chapter 2 Class 9 | Polynomials, Factor Theorem Questions | NCERT Maths

Factorisation of Polynomials by SPLITTING THE MIDDLE TERM Explained| Class 9 |Chapter 2| NCERT Maths

Example 8 Chapter 2 Class 9 | Polynomials | 1St Solution | Splitting Middle Term Questions | Maths

Example 8 Chapter 2 Class 9 | Polynomials | 2nd Solution | Factor Theorem Questions |NCERT Maths

Example 9 Chapter 2 Class 9 | Polynomials | Factorise using Factor Theorem Questions |NCERT Maths

Example 10 Chapter 2 Class 9 | Polynomials | Factorise cubic or degree 3 polynomials | NCERT Maths

Question 1 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 5 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

EASY TIPS & TRICKS 🔥 TO LEARN ALGEBRAIC IDENTITIES | Class 9 | Chapter 2 | Polynomials | NCERT Maths

Example 11 Chapter 2 Class 9 | Polynomials | Product of polynomials using algebraic identities

Example 12 Chapter 2 Class 9 | Polynomials| Example 17 |Product of big numbers without multiplying

Example 13 Chapter 2 Class 9 | Polynomials | Factorise polynomials using algebra identities | NCERT

Example 14 Chapter 2 Class 9 | Polynomials | Expand polynomials using algebraic identities | NCERT

Example 15 Chapter 2 Class 9 | Polynomials | How to expand polynomials using algebraic formulae |

Example 16 Chapter 2 Class 9 | Polynomials | Factorise polynomials using algebraic identities

Example 17 Chapter 2 Class 9 | Polynomials | How to expand polynomials using algebraic formulae

Example 18 Chapter 2 Class 9 | Polynomials | Example 23 | Cubes of big numbers without multiplying

Example 19 Chapter 2 Class 9 | Polynomials |Factorise cubic polynomials using algebraic identities

Example 20 Chapter 2 Class 9 | Polynomials | Factorise cubic polynomials using algebraic identities

Question 1 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 5 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 6 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 7 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 8 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 9 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 10 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 11 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Class 9 Chapter 3 | Coordinate Geometry

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State and Prove Remainder Theorem – Polynomials

Sunil Syal — Tue, 15 Sep 2020 06:39:56 +0000

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.

Watch on YouTube

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:

It will be divided completely and the remainder will be 0
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

The post State and Prove Remainder Theorem – Polynomials appeared first on Mathemafia.

remainder Archives – Mathemafia

Class 9 | Chapter 2 | Polynomials | Factor theorem

Prerequisite

Chapter 2 – Polynomials

Theorems / Concepts

Exercise 2.1

Exercise 2.2

Exercise 2.3

Exercise 2.4

Exercise 2.5

Exercise 2.5

Examples

Examples

Examples

Good to learn

State and Prove Remainder Theorem – Polynomials

Remainder Theorem

Proof

Why Remainder is a constant?

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

Final Deal 💰

Prerequisite