Sunday, February 14, 2021

EFFECTIVE COMMUNICATION AND POSITIVE PERSONALITY WITH MINDSET

 

BETTER PERSONALITY WITH EFECTIVE COMMUNICATION AND PROGRESSIVE MINDSET


POSITIVE PERSONALITY
        Here we need to discuss as how we can find success in our real life and at our professional/social environment.
All we work with passion but we find we not find much success as we wish ; that due to our work plane and our self confidence. If we have good personality and our communication skill good then we can attract the attention of any one to do work according to as we need to do with them. For better growth progressive mindset have big role in our real life success.

1. How problem solving helps individual in personality development and growth?


        We know well that all our body behaviour / impressions when we speak or when we act depends on our confidence. When we think as we can talk about some topic and have confidence ; then that time our body language will already seems as a strong personality; but if we think weak us in respective topic and think as we not have much knowledge about topic then automatically our body find downfall in impressions and we seems us weak and nervous to talk with someone or in group. Our eyes impressions or body language depends on our think and our confidence between conversations.
      For above reasons its important we have full knowledge about topic and have extra knowledge about all subject which can be use during conversations.

      To gain knowledge not a simple method that can be find so sudden that need we work continuously about gain information and work on rectify according to our use. We need to learn much more. Need to watch our surroundings / social groups or social medias and share our thoughts between them and have relationship between them which have same type positive thoughts and can be grow our skills or knowledge.

        If we find difficulties then we can solve problems by understand problems and work accordingly.
We can help others to solve their problems and we can guide them to find solutions.
That way we can improve personality or respect between them.

        Never need to blame to any others if you not success. Have confidence on your self and learn about situations. Situations can be give us solutions if we watch carefully and try to find reasons about problems. 
Never loose your confidence and try to work on problems and follow progressive mind set to find solutions by adopt learning and experimental attitude.
Book or social media/ google search or other search sides have many ideas which can guide us right direction to find solutions.

Its a fact if we help others then we can find confidence and we can grow our reputations to. 
No any can respect you if you not able to help them when they need your guide or any other help which you can. Better we respect them and try to work with them to help in find solutions. 
That practice can be improve your knowledge or strength . We can rise positive peoples circle or groups by help them unconditional. That way they can be find impressions on your skills or attitude and all that results as growth in your reputation or personality. That all give you confidence to work much better and find skills to share your knowledge or skills between peoples.


2.Personality 


Body behaviour / Impressions
Personality is a characteristic inside ourselves, which we keep from birth. Our family environment, their ways of living, our family behaviour, emotions and their values ​​have a lot of influence on personality. Our gestures, our manner of speech, our dress, our attractiveness, character, our emotions, our thoughts, abilities, interests, attitude, behaviour, our attitude, our way of trying and our confidence. Our personality is the main identity.

       Our personality can never match with others, because everyone's culture and family environment and promotions are never the same. These characteristics give us a different identity (unik) from others.

No person is fulfilled by birth, we can bring change in our positive thinking, by believing in ourselves and with confidence. No matter how bad our family circumstances and environment are, we can achieve success, we can also get attractive personality. We just have to believe in ourselves and we have to make continuous efforts.

Our personality is of two types; External personality and internal personality. Our external appearance, our way of life, our body language, our ways of talking, our way of walking; Are reflective of our external personality. Our emotions, our perspective of seeing and thinking, positive attitude , confidence, inner thinking, self-realization, our way of thinking, character, moral behavior, our sense of self-respect, our responses etc. reflect our inner personality .
Do not depend on others, trust in yourself and your efforts. Avoid as much debate and conflict of interest. Do not insult anyone nor tolerate insult to anyone, if anyone dares to insult you, give him a quick reply; So that he should never dare again. Never climb anyone's head nor let anyone climb on your head. Always be with talented people, keep distance from the people who are barbaric and courageous at the time of failure. Never be afraid of troubles; Confront them with confidence and skill. Stay away from fear in future. Never think of your work as small, make it big on the strength of hard work. Whatever be the situation, always be happy and keep smiling on your face. Always speak thoughtfully whatever you say, as far as possible speak less and listen more to the other.
Your way of speaking should be decent; Never falter while speaking, speak patiently and keep smiling on your face; Talk with your eyes with confidence. The brightness of the eyes should always remain, the lofty man too. Let's go with decency and confidence. While walking, your face is full of confidence and straight, always walk in front or down; Excessive soldering - do not go looking; This will enhance your personality. Do not look at someone while walking, nor make any comments; This will adversely affect your personality. Laugh softly, do not let your personality get depleted by loud or loud; One should always laugh only after seeing the atmosphere and situation. Whenever you meet someone for the first time, speak less, speak slowly and politely, keep a distance, ask what you ask with love, answer what is asked in front of you; This will make people eager to meet you, respect you. A positive smile should always remain on the face throughout the conversation. Always the front should feel special.
If you have to show resentment, show it in silence, if the person in front knows to respect you; So he will understand everything - with your silence. See the front and react. Never be immediately available, so that they feel you depend on their grace; It is in your hands to maintain your self-esteem and self-pride. Respect others. Give importance to the words of others. Keep your way of thinking positive. Keep your gestures positive and courteous. Your dress should be attractive and elegant; Choose clothes which suitable for work and situation.

   Your attitude should be positive. Positive approach gives us the motivation to move in the right direction with confidence, our personality will be positive. A positive attitude is necessary for happiness and progress in life.

      When our thinking is negative then the probability of failure increases, we start doubting our abilities, our confidence also starts to decrease. Therefore, for the success of work, negative thoughts have to be controlled. We can change our attitude and change our life, we can be successful. To do your best, you have to turn negative thinking into positive, you have to be confident - on yourself and on your ability.



POSITIVE ATTTITUDE

Following are some things, if we keep in mind, we can achieve all successes and happiness in life by bringing changes in our personality, positive attitude: -



1. You should respect the right things of the younger ones and accept the advice of the elders. Do not get angry and quarrel over small things and consider the reasons. Treat people the same way you want for yourself from others, only then you can get respect. Be polite, bring a glow of confidence and smile on your face, introduce civilization in conversation. Do not display ego (EGO) too much.

2. You should make an effort to be special to others. Do not look at others, bring positive changes in yourself. Your way of life, mode of conversation, your personality should be different from others. If someone is giving you respect and respect, then you too become theirs and honor them wholeheartedly. Do not humiliate and ignore them. You can win their hearts with your good behavior and faith.
 All these will increase your respect and self-pride.

3. Do not do any work in a hurry. Work only after understanding the success of the work and their good and evil. The success of every task demands time and dedication, so be patient and patient, till the success of the work.

4. Be honest but do not let it be used. Love but do not hurt yourself, believe but do not be naive. Do not lose your voice to listen to others.

5. Never fall prey to anyone's tricks.

6. Never give suggestions without asking anyone. If someone does not show interest in talking to you, then leave it. Realize your importance.

7. Trust yourself and your ability; Never be dependent on others. Keep positive thinking and be confident in yourself. Don't make anyone special nor be someone special. Keep as few special friends as possible.

8. Stay away from habitual conflicts and conflicts; With this your time will be disputed and your attention will deviate from positive actions and disturbance will be born in the mind. Remove the solution from the conversation, not from the vehicle.

9. Make yourself mentally powerful. Keep positive thinking about life and success. Stop calling yourself successful; Your success should be seen by your work and your enthusiasm; not that you have to tell. Only, you need to make yourself worthy, the world will follow you. You do not have to stop on your path of progress, you have to move forward.

10. You do not have to pay attention to your past, focus on the present, think about the goals of your successes, do not be discouraged by thinking about the unwanted problems that will come. If there are problems then you will  also resolve with confidence; Don't get frustrated and change your goals.

11. Do not burn on the successes of others, congratulate them on their successes, if possible, help them as well. By doing this they will also become yours, also helpful in your success.

12. Never blame anyone else for your failures. Never compare yourself to others, what you have to do is to do it on the strength of your confidence and dedication and hard work. Seek your happiness and success in yourself and not in others.

13. Don't waste time. Focus on work, be punctual. Never be too enthusiastic, go ahead with positive thinking. Have complete confidence in yourself.

14. Learn from failures with positive thoughts. Do not copy the crowds, do the mind.

15. Always speak less and listen more to others, this will give you a lot to learn and your good personality will affect others.

16. Focus on the development of your personality. Your manner of speech, physical expressions should be polite and full of confidence and enthusiasm. Speak with your eyes in conversation, your facial expressions should be cheerful and attractive. Speak with emotions.



3. Communication Skill 


                       Communication

                                                                      
      Conversation means speaking or saying something. We use many mediums to say or reveal something; For example, by speaking, writing, or using any medium of communication, we can convey our words to anyone. The art of conversation has a great importance in achieving our success. In moving forward in any field, of business. Success also depends on the art of conversation (communication); If we are not able to impress our customer with our words, then we can never succeed in business.

      The importance of all our relationships also depends a lot on our behavior and colloquial methods and practices. A single word is a precious thing that can strengthen a relationship, and in a moment can ruin a relationship. A word that comes out of our mouth can make or break a relationship in a moment. This is the art by which we can become a well-successful businessman, speaker, administrator. Each word that comes out of our mouth is very important. In the conversation, we must speak confidently, respond and seek answers.

If we are sociable and know how to speak well, then everyone will be impressed by us and our colleagues too. We can make anyone a friend or even an enemy by our speech and behavior. A good administrator or businessman is the one who is a good speaker and is successful in speaking to the next person and understanding the situation and can expose his hidden problems and doubts from the front.


      Voice is the words that express love or hate; Just the way of speaking will be different, results will be obtained accordingly.

     Remove this fear that we do not know how to speak effectively, if we have never spoken on a particular topic in a group, how will we be able to speak; What will people say about our shortcomings; These are all those fears which try to keep us away from speaking in an effective way and reduce our confidence. No one has all the characteristics or abilities since childhood, only by our efforts in positive direction can we overcome our hesitation and shortcomings. For this, we have to believe in ourselves and our abilities and make efforts with confidence and we have to increase knowledge related to the subject only then we can become good speakers. Fear of some stage is not bad, it will serve to inspire us to learn; But by accepting this fear, we have to plan with confidence and make it successful. Whenever you speak, speak confidently with a smile on your face. There should be power in your speech and you should speak each word clearly and thoughtfully. It is necessary to have sweetness and reflex in speech. Whatever you talk, see the time and the peace so that people will feel honored for you by your words; Not that they feel angry or sad.

     Hearing your conversation, those in front of you should be interested in you and not feel bored after listening to you. Whatever you say, speak with ease and a loving smile; If possible, include the person in front of you. Also keep asking him your feedback or suggestions and questions, so that he feels that you are interested in him. Do not be afraid to say anything. Whatever you say, speak with confidence. While talking, talk with eyes contact in them, this will increase your confidence and the next person will like to talk to you. While speaking, the physical gestures should be attractive and confident, the dress should be matching with the environment and situations and the subject. While talking, your hand gestures and rhythms should reveal depth in your words. Talk to the front and understand. If someone does not agree with you, do not get entangled with it, nor argue with it; Give others a chance to speak as well.

Whatever you say, focus on the subject itself, distracting the attention by speaking out of the subject will disturb the seriousness of the subject and we will not be able to leave our influence. While speaking, ask the person with sweetness and confidence, this will give you a chance to know him and by knowing his thinking and willpower, you will be able to put your words in an effective way. Try to make good sense from the front, show smile and sadness according to their mood. Give good feelings to others. Your feelings should be good and cooperative towards others. Do not reveal your pride in things in front of others. Those ahead will respect you and your words wholeheartedly. While speaking, do not let the reactions or thoughts of the people dominate you, keep a cool mind and respond with positive smile. Keep your confidence up.

These are the main things that can make our conversation effective.

Following are some more things from which you can improve the art of conversation (communication) more thoroughly. With this you can gain respect and charm in conversation: -

1. Speak boldly and confidently. You should have a grasp on the subject and your voice should be attractive, sweet and powerful.

2. Your body language and attire should be attractive and positive. The gestures of your hands, eyes and body should be in harmony with the subject and environment of conversations .

3. Along with speaking you should also listen to the people in front. The likes of those in front should be considered as disliked. As far as possible, include the people in the front and learn to say in brief.

4. Get complete information about your subject and try to read the frontal mind well before speaking. You speak less, give the person in front a chance to speak more; With this you will be able to check his likes and dislikes and his thinking. Increase your vocabulary so that you can increase your influence by using effective words. Whatever you speak, choose the words and speak and keep the language decent. You have to speak politely and sweetly with confidence and you should have strength in speech. Avoid using any obscene and offensive words.

5. You should come to see situations and talk; cleverness in talking should also be used according to situations. If you want to say something harsh, it would be better to speak SORRY before speaking. Speak less and hear more of the hair in front; Then, if necessary, give your response with sweetness and keep your point. It is also good to keep quiet at times; Do not forget to give advice without asking. Avoid making negative comments while talking, never cheat anyone; They will increase your respect in the eyes of those in front of them and they will be willing to take interest in your talk. Positive smile in things, praising someone on their good work or their success will help you in increasing your respect. Do not fall in love with others and learn to agree with their words; Talk to people looking forward to their feelings and mood. If the next ones are talking, then keep giving them the expression of their consent by saying gestures or yes-yes etc. in a nutshell; This will reveal your affinity to those in front.


4. Mindset :



Leadership
          Mindset has special importance in our life. We know that our mind starts working according to the way we think. Our feelings of mind are determined by our way of thinking. You must have noticed that when we have an energetic speech If we listen, for a moment we also find ourselves full of enthusiasm and powerful thoughts start coming in our mind. The day we start a day with complete preparation, then on that day we automatically Find yourself happy and exhausted; But as soon as a task is not successful in our mind, we find ourselves tired and frustrated at times. All these show our mindset.

          From leadership to love mentality determines our every aspect. We work as hard as we can. We should have full confidence in our abilities; The business must tailor the work to the condition; Only then possible success can be achieved. Belief  has a lot of power, which has a great impact on our conscious and subconscious mind. Believe in yourself, nothing is impossible. If we, with positive thinking, persevere. - If we work hard with certainty, success will definitely be achieved at a certain time. We have to mold our personality, faith and perseverance according to the working condition.

         There are two ways of thinking - one is Positive / Growth / Progressive Mindset and the other is Negative / Fixed Mindset.

          In positive thinking, no action is impossible, if we have passion and faith - to do the hardest thing; And be obsessed - to fight hardships. Positive thinking people never trust fate; They believe that every task will be successful only when we work hard and can be made big by doing small tasks. They do not believe in innate intelligence. They believe that we can work on our strengths and Knowledge can be increased and all achievements can be achieved by learning. Character, intelligence and creative ability of Positive / Growth / Progressive Mindset grows according to work, as they do not panic and persevere. With hard work to achieve the task by earning knowledge with hard work and succeed. I am always ready to face every challenge with positive thinking. They consider failure as a new challenge with a new thinking. They open their doors of development and emphasize on new creation and efforts. They make positive efforts and positive thoughts with positive thinking and succeed. They believe that even though we have lesser qualities, Still we can achieve success through untiring effort and continuous learning These are the challenges, they face the problems with bravery and patience. Due to positive thinking, they do not read any effect of the criticisms; Never get upset seeing problems, they never get jealous of those who are with them, nor do they ever make any allegations against them. They give them success in the success of others and accept everyone's support in their success. Is that through hard work intelligence can be improved by development in personality. They mold themselves according to work. They know to be patient, hard work is their pre-determined goal; They never panic by learning.

            Whereas Negative / Fixed Mindset believes in innate intelligence and is afraid of losing and criticized, causing such people to be afraid to take risks and try to hide their shortcomings; As a result, most face defeat. The character, intelligence and creative ability of the Negative / Fixed Mindset is fixed, as they are afraid of losing and learning to maintain their image and doing little or little work. Their ego (ego) and fear dominate them.

          Therefore, if you want to be successful and enhance your personality, then think positively and live in a positive atmosphere, abandon the laziness. No success is impossible; If our intentions are firm and we make continuous efforts in the right direction, with the right information. Great effort will have to be made in doing big things, results will also get big. Quit making excuses.

          Recognize your abilities and characteristics, have a victorious personality. Always think that no action is impossible; If we try continuously, we will surely get success. Keep creative ideas and find new ways - to make the work easier. Find the reason of failure and start taking positive measures and efforts.

          Never depend on others, find your own way and keep your intentions . Stay away from negative thinking. Always meet positive thinking people and respect them. Never be afraid to take risks and try success with positive efforts. Remove Fear , face the fear with confidence. Stand out from the crowd, never copy anyone; Choose your path with creative thought. Don't make excuses and run away from your goal; Brave the problems bravely. Try your best before you give up and assess the situation; Surely some new way can be found.

         To achieve the goal, you have to work hard with dedication and strong will. Keep an open mind.

         Knowledge can be bought, just you should come to use it. Develop your imagination and have full confidence in yourself.

         Do not waste time looking for luck and opportunities, start the work as quickly as possible. Make the plan practical and complete. Co-ordinate with colleagues.




MEDITATION AND ITS IMPORTENCY IN REAL LIFE SUCCESS OR HEALTH ISSUES. FIND TIPS FOR MEDITATION IN CORECT WAY.

 

MEDITATION


   The entire control of our body is under the control of the brain. The control of the brain is through the mind; All control messages go to the entire system from here.

       Meditation is the best way to clear the mind. Meditation also has a positive effect on our subconscious mind, the combined effect of which we can achieve many amazing solutions using positive-speaking (AFFIRMATION). An energy cycle is formed around the body; In which all the diseases, defects of the body are removed by the use of the created world power, there is a feeling of new zeal, new energy, consciousness, enthusiasm in the body. We get wisdom and knowledge.


        Meditation has the same meaning in all religions; The aim of which is - for the purification of the body, the soul-air enters the body and fulfills concentration, enthusiasm.



1. BASIC STATE OF MEDITATION:

In the initial state of meditation there are the following main stages: -

1. Waking up early in the morning and taking a deep breath.
    This will reduce sleep; But sleep will be deep and it will feel happy.
2. Take food lightly.
3. Maintain concentration on the breath.
4. Fast the senses:
    Stay silent / keep your eyes closed / keep your ears closed.
    These actions will increase the power, focus will be concentrated.
5. Be happy during the whole time of meditation, laugh, be excited and happy.

If we are proficient in the above, we can practice meditation.

2. MAIN STAGES OF MEDITATION :

In this, the practice is completed in three stages : -


1. Sense of dedication - Take deep breaths, concentrate full attention on your breath. (Do this action for 10 minutes)

2. Sense of acceptance - all things, circumstances, environments are like; Similarly, accept and meditate (do this action for 10 minutes).

3. The sense of death (erasure, lost and lost) - keep expressions while doing meditation in it; As you have disappeared, you are experiencing a new world. This will make you feel free from bondage, trouble-free, happiness, love. (Do this action 10 minutes)

      After sitting in the three steps above, sit on a clean pedestal and perform the above actions every 30 minutes after waking up in the morning and 30 minutes before going to bed at night in a clean secluded environment.


Meditation is a technique whose success depends on your faith and passion.


3.Benefits of meditation 


     Meditation makes the mind feel lighter. There is amazing growth in concentration, confidence, positive thinking, memory, health, will power. Meditation is an attempt to bring the reality of nature.

1. Meditation changes the physical and mental state of the brain. Meditation increases the size of the positive part of the brain and reduces the size of the negative part. This has the effect that we feel more energized, focused. It increases self-confidence, ability to think and understand. There is an increase in positive thinking. Concentration increases.

2. Meditation leads to amazing growth in mental health, positive buildings are strengthened. Relieves from fear, anxiety, stress. The feelings of love are awake, the impulse of anger decreases. We get new energy, power, There is a feeling of self-confidence. The environment seems to be fearless. Will power increases with attention; We can also reduce bad habits.

3. Meditation controls the amount of METABOLISM in the body; As a result of which we can increase or decrease our load (OVER-WEIGHT / UNDER- WEIGHT).

4. Meditation reduces the amount of C-REACTIVE PROTEIN present in the blood; As a result, it is possible to treat high blood pressure (HIGH BP), many cardiovascular diseases. CARTISOL will be low; Which will relieve stress, will also give relief in sleep.

5. Meditation has positive effects on the skin; SLOW AGING,  increases facial glow, happiness will increase .




I look forward to your valuable suggestions and ideas. Please give me support to improve my efforts to give valuable knowledge among peoples.



MATHEMATICS IN EASY STEPS : Polynomials and Linear Equation (Definition, types and methods of solutions with examples for better understanding.)

 

  "Mathematics is a easy subject but maximum find them so hard to learn and remember Mathematics functions and rules. That all because we not give time to understand basics of Mathematics rule. So learn here the basic rules step by step"

I try to explain all topics in easy tips or steps so a learner can easily understand basic of mathematics. I try to give much knowledge in short notes. All explain with necessary examples, I sure if you read well my topic and try to solve all examples which I give here, then you can solve any problem without any teacher.

(A), POLYNOMIALS


Polynomials are algebraic expressions that consist of Variables , Coefficients , and Constant .

We can do easily addition , subtraction , multiplication , division and also exponential operations on polynomials but never be division by variables.

        Standard form of polynomial is :  P(X) = aX2+bX+c

                    Where P(X) = Given Polynomial,

                                 a, b = Coefficients,

                                  C = Constant Value,

                                 X, Y = Variables,

                                 In XX ; 2 and 1 are exponents.

                                and + or - symbol called Operators.

        Example :  2, 5, √2 , √5 , 21 , 35 , 5∕∕2 X , 12X + 5 XY + 6Z+0.5 are Polynomials.

                               (BUT)  1/X , (1/X) +3 , X−2 , √X ,  3X−2,  2/ (X+5) are not Polynomials.

NOTE :  All numbers are polynomials because we can arrenge numbers in polynomial form.

Example : for number 2 , we can arrenge as ;  0X2 + 0X + 2 etc.


1. CLASSIFICATION OF POLYNOMIAL :

        There are 3 types of Polynomials which are ;-

            1. Monomial

            2. Binomial

            3. Trinomial


1. Monomial :

    That type polynomial consist only one term .

    EXAMPLE : 2X , 5 , -2XY , 6Z2,6X2 ,  etc.

2. Binomial :

    That type polynomial consist two terms .

    EXAMPLE : 2X+3 , -5X+7 , 6X2+2X ,  6X2Y+XY , etc.

3. Trinomial :

    That type polynomial consist three terms .

    EXAMPLE :  6X2- 2X+3 , 6X2-5X+7 ,  etc.


2. DEGREE OF POLYNOMIAL :

        " Degree of Polynomial is a highest degree of a monomial in a polynomial. "

We can classify polynomial accordind to degree of polynomial , such as -

(a). ZERO POLYNOMIAL : Here degree of polynomial is zero (0).

                                                  Example : P(X) = 9

(b). LINEAR POLYNOMIAL : Here degree of polynomial is ONE (1).

                                                  Example : P(X) = 9X + 5

(c). QUADRATIC POLYNOMIAL : Here degree of polynomial is two (2).

                                                  Example : P(X) = 9X2 + 5X + 2

(d). CUBIC POLYNOMIAL : Here degree of polynomial is three (3).

                                                  Example : P(X) =  5X3 + 9X2 + 5X + 2

(e). QUARTIC POLYNOMIAL : Here degree of polynomial is four (4).

                                                  Example : P(X) =   5X4+ 5X3 + 9X2 + 5X + 2


3. TERMS OF POLYNOMIAL :

     The parts of polynomial which separated by sign " + ' and " - " called terms of polynomial.

       If  P (X) = 9X2 + 5X - 2 then ,

            Degree of polynomial = 2 ,

            Terms of polynomial = 3 (As  9X2 , 5X , 2)


4. DIVISIBILITY RULE AND FACTOR THEOREM :

        " If polynomial is P(X) , then P(X) is divisible by a binomial ( X - a) if and only P(a) = 0  " 

If in polynomial P(X) ; P(a) = 0 then ( X - a) is a factor of given polynomial (OR) if P(-a) =0 ; then ( X + a) is a factor of given polynomial.


5. ZERO OF THE POLYNOMIAL P(X) :

        If in polynomial  P(X) = aX2+bX+c  ;

        If P(m) = 0 , then m is a Zero of Polynomial of the given polynomial.

    EXAMPLE :  

        If P(X) = 5X2-5

        Here,  P(1) = 5( 1 )2-5 = 5 - 5 = 0

        ⇒ P(1) = 0

    Hence 1 is a Zero of polynomial P(X).

NOTE : 1. Every linear polynomial has one and only one Zero.

                2. A polynomial can have more than one Zero.

                3. " 0 " may also be a Zero of a polynomial.


6. REMAINDER THEOREM :

        " If a polynomial P(X) is divided by (X - a) ; then the remainder will be P(a) . "

        If in polynomial P(X) if P(a) = 0 ; then is a Zero of polynomial , but if  P(a≠ 0 and 

give some value then that value called remainder.


EXAMPLE 1 : 

    If P(X) = 3X+ X - 1

    Here , P( -1 ) = 3( -1 )+ ( -1 ) - 1  = 3 - 1 -1  = 1

Here , 1 is remainder and  ( 3X+ X - 1 ) is ' dividend ' , ( X + 1 ) will be ' devisor '.

            DIVIDEND = ( DIVISOR  x QUOTIENT ) + REMAINDER   

By divide method we can understand that problem well ; such as above equation :--


X + 1 )  3X+ X - 1  ( 3X - 2  ← ( QUOTIENT )

              3X+ 3X 

             -        -

         ________________

                       - 2X  - 1

                       - 2X  - 2

                       +       +

               _________________

                                   1  ( REMAINDER )


EXAMPLE 2 :

Find the remainder if   2X4X3 - X2 + 3X + 1  is divided by X + 1 .

SOLUTION : 

    P(X) =  2X4X3 - X2 + 3X + 1

Here ,  Zero of  ( X + 1 )  is  ( - 1 )

Hence , 

 P( -1 ) =  2( -1 )4+ ( -1 )3 - ( -1 )2 + 3( -1 ) + 1

               = 2 - 1 -1 -3 + 1  =   -2

Hence , -2 is the remainder when 2X4X3 - X2 + 3X + 1 is divided by X + 1 .

NOTE : If in polynomial P(X) ; if P(a) = 0 ; then  (X - a) will be the factor of the given polynomial.

( Since X = a ;  so that  X - a = 0 ) 


7. FACTORISATION :

    FACTOR :  

        " If an algebraic ex[pression can be written as the product of two or more expressions , then the each of these expressions are called Factor of the given expression. "

    FACTORISATION 

       "To divide an expression in two or more small expressions called factorisation . Each small expression will be called the factor of given expression."

    FACTOR THEOREM

          " If in a polynomial P(x) , "a" is any real number,then if P(a) = 0 ; then (x-a) is a factor of given polynomial P(x). " 


7(1). METHODS OF FACTORISATION :

        (A). BY TAKE COMMON FACTOR :

                EXAMPLE 1. : 

                 5x2y + 125 xy2 - 25 x3y2

                = 5xy ( x + 25 y - 5x2y )

                 EXAMPLE 2. : 

                 28 ( 5x + 9y )2 + 21( 5x + 9y )

             = 7 (5x + 9y ) ⟨ 4 (5x + 9y )+3(1)⟩
             = 7 (5x + 9y ) ( 20x + 36y + 3 )

         (B). FACTORISATION BY GROUPING :

                If there possible to arrange given expression in suitable groups in which groups have a common factor , then we can solve expression by use common factor method after arrange in group. 

EXAMPLE :

            4ax2 + 6by2 + 2bx2 + 12ay2
            ⇒ 4ax2 + 2bx+ 6by2  + 12ay2
               ⇒ 2x2 (2a+b) + 6y2(b+2a)
            ⇒ (2a+b) (2x2+6y2)
            ⇒ (2a+b) 2 (x2+3y2
            ⇒ 2 (2a+b) (x2+3y2)

           (C). BY USE SPECIAL IDENTITIES :

    Following are main identities which can be use in factorisation of given expression :-

        1. (a+b)(a-b) = a2-b2
           2. (a+b)2  = a2+2ab+b2
           3. (a-b)2 = a2-2ab+b2

EXAMPLE :

1.  25 - 16x2  
     = (5)2-(4x)2
    By use a2-b2 = (a+b)(a-b)  ; we get-
    = (5+4x)(5-4x)

2. 75a2-48b2 = 3(25a2-16b2)
    = 3⟨(5a)2-(4b)2⟩ 
    By use a2-b2 = (a+b)(a-b)  ; we get-
    = 3(5a+4b)(5a-4b)

3. 16(a+b)2-25(a-2b)2
    ⟨4(a+b)⟩2-⟨5(a-2b)⟩2
    By use a2-b2 = (a+b)(a-b)  ; we get-
    = ⟨4(a+b)+5(a-2b)⟩ ⟨4(a+b)-5(a-2b)
    = (4a+4b+5a-10b) (4a+4b-5a+10b)
    = (9a-6b) (14b-a)

4. 16x2-4y2+20y-25
     =16x2- (4y2-20y+25)
     By use identity (x-y)2 = x2-2xy+y2  ; we get -
     = (4x)2 - (2y-5)2
      By use a2-b2 = (a+b)(a-b)  ; we get-
     = 4x+(2y-5)⟩ ⟨4x-(2y-5)
     = (4x+2y-5)(4x-2y+5)

5. Evalute : (410)2-(390)2
        (410)2-(390)2
        = (400+10)2-(400-10)2
        = ⟨(400+10)+(400-10)⟩ (400+10)-(400-10)
        = 800 x 20 = 16000

6. 25x2+10x+1
    = (5x)2+2(5x)(1)+(1)2
    By use identity (x+y)2 = x2+2xy+y2  ; we get - 
    = (5x+1)2

     (D ) SOME SPECIAL EXAMPLES ON FACTORISATION OF POLYNIMIALS :

EXAMPLE 1.  If x+1/x = 5 then find the value of -
        (1). x2+1/x2    (2). x4+1/x4

SOLUTION :    
        (1).  (x+1/x) = 5
                ⇒ (x+1/x)2 = (5)2
                ⇒ x2+1/x2+2(x)(1/x) = 25
                ⇒ x2+1/x2+2= 25
                ⇒ x2+1/x2= 23

        (2). (X2+1/X2)2 = (23)2
                ⇒ X4+1/X4+2X2x1/X2 = 529
                ⇒ X4+1/X4 = 527

EXAMPLE 2.  If x-1/x = 5 then find the value of -
        (1). x2+1/x2    (2). x4+1/x4

SOLUTION :    
          (1).  (X-1/X)2 =(5)2
            ⇒X2+1/X2-2Xx1/X = 25
            ⇒ X2+1/X2-2= 25
            ⇒ X2+1/X2 = 27

         (2). (X2+1/X2)2 =(27)2
            ⇒X4+1/X4+2X2x1/X2 = 729
            ⇒ X4+1/X4= 727

EXAMPLE 3.  (2X+3)(2X-3)(4X2+9)
SOLUTION : (2X+3)(2X-3)(4X2+9)
                ⇒ (4X2-9)(4X2+9)   ( By use identity : (a+b)(a-b) = a2-b)
                ⇒ (4X2)2-(9)2     ( By use identity : (a+b)(a-b) = a2-b)
                ⇒ 16x4-81

EXAMPLE 4. If x+y = 15 , xy =20. Find the value of (x2+y2)
SOLUTION : 
            Here , x+y = 15
            Hence, (x+y)2 =(15)2
            ⇒x2+y2+2xy = 225
            ⇒x2+y2+2(20) = 225
            ⇒x2+y2= 225 -40
            ⇒x2+y2= 185

EXAMPLE 5. If x-y = 15 , xy =20. Find the value of (x2+y2)
SOLUTION : 
            Here , x-y = 15
            Hence, (x-y)2 =(15)2
            ⇒x2+y2-2xy = 225
            ⇒x2+y2-2(20) = 225
            ⇒x2+y2= 225 +40
            ⇒x2+y2= 265

EXAMPLE 6. Examine that X+1 is a factor of 3x2+x-2
SOLUTION : 
                    Here zero of x+1 is -1.
                    Let,  P(x) = 3x2+x-2
                        ⇒P(-1) = 3(-1)2+(-1)-2
                        ⇒P(-1) = 3-1-2 = 0
So, by the factor theorem, (x+1) is a factor of given polynomial 3x2+x-2

EXAMPLE 7. Find the value of K if (x+1) is a factor of 5x4+2x3-4x+K
SOLUTION : 
                    Here P(x) = 5x4+2x3-4x+K
                    Since (x+1) is a factor of P(x), hence 
                    P(-1) = 0
                    ⇒P(x) =5x4+2x3-4x+K
                    ⇒P(-1) = 5(-1)4+2(-1)3-4(-1)+K = 0
                    ⇒5-2+4+K = 0
                    ⇒7+K = 0
                    ⇒ K = -7

EXAMPLE 7. Find the value of x if equation is 5x-15
SOLUTION : 

( Method: To solve polynomial equations we need to take "0" at right hand sides of equation; then use basic algebraic methods and factorisation concepts to solve equation.)

        P(x) = 5x-15
              ⇒ 5x-15 = 0
              5x=15
              ⇒ x = 15/5 = 3
        Hence, 
                 x=3

EXAMPLE 8. Find the value of x if equation is  3x2-x+x3-3.
SOLUTION : 

    ( Method: To solve polynomial equations we need to arrenge the equation in descending order of degree of coefficients then equal that equation as zero.)

        3x2-x+x3-3
        ⇒ x3+3x2-x-3=0
        ⇒ x2(x+3)-1(x+3)=0
        ⇒ (x2-1)(x+3)=0
    Hence,
        If  x2-1=0
            ⇒ x= 1
            ⇒ x = 1, -1    and,
        If  x+3 = 0
            ⇒ x = -3
    Hence,
        Solutions are = 1, -1, -3

EXAMPLE 9. Find the value of Polynomial as indicated the value of variable:-
    (1). P(x) = 6x2-5x+5  ,at x=2
    (2). P(m) = 4m4+5m3-2m2+2  ,at m=a

SOLUTION :
    (1). P(x) = 6x2-5x+5 
           ⇒ P(2) = 6(2)2-5(2)+5
                     = 24-10+5 = 19
    (2). P(m) = 4m4+5m3-2m2+2
            ⇒ 4(a)4+5(a)3-2(a)2+2
            ⇒ 4a4+5a3-2a2+2
                 



7(2). FACTORISATION OF QUADRATIC EXPRESSIONS :

    Here the expression will be in the following form :-
        ax2+bx+c
      For factorisation we need to divide middle term " bx " in two parts such as which product (multiply) equals to (axc) and the sum should be eqyals to " b " .

EXAMPLE 1. Find the value of x for the expression 2x2+11x+15 .
SOLUTION : P(X) = 2x2+11x+15
        Here the product of first and last term's coefficients = 2x15 =30
        Now we need to divide 30 in two parts , which sum = (+11)
        Here we get 6+5=11, 6x5=30
Hence,
    2x2+11x+15 = 2x2+6x+5x+15
    ⇒ 2x(x+3)+5(x+3)
   ⇒ (x+3)(2x+5)

EXAMPLE 2. Find the value of x for the expression 2x2-x-15 .
SOLUTION : P(X) = 2x2-x-15
Here the product of first and last term's coefficients = 2x15 =30
        Now we need to divide 30 in two parts , which sum = (-1)
        Here we get -6+5=-11, -6x5=-30
Hence,
    2x2-x-15 = 2x2-6x+5x-15
    ⇒ 2x(x-3)+5(x-3)
    ⇒ (x-3)(2x+5)


(B). LINEAR EQUATIONS IN ONE VARIABLE

1. EQUATION : 
        A statement of equality of two algebraic expressions which have or more variables is called equation.
2. LINEAR EQUATION :
    If any equation having any linear polynomials is called linear equation.
In other words " A mathematical statement that has an equal to the "=" symbol is called an equation. Linear equation are of degree 1."
EXAMPLES :
            2X+4 = 9X
            2(X+2) = 5
            1/3 X -2 = 2X+9

3. SOLUTION OF LINEAR EQUATION IN ONE VARIABLE :
    We can understand the method of solve Linear equations in one variable by these examples :--

EXAMPLE 1. Find the value of x for the expression 2x+4=9x-7 .
SOLUTION :
        2x+4=9x-7
We need to carry all x value in one sides and numbers in other side. By solve that result ,we can find unknown value of x .
(NOTE : When we change sides of a given term then sign will be change.) 
        ⇒ 2x-9x= -7-4
        ⇒ -7x = -11
        ⇒ x = -11/(-7) = 11/7
        Hence, x = 11/7

EXAMPLE 2. Find the value of x for the expression :-
                    (x+5)/4 + (x-3)/5 = (5x-8)/2
SOLUTION :
(x+5)/4 + (x-3)/5 = (5x-8)/2
Here LCM of 4,5,2 is 20. So, multiply equation by 20 both sides we get-
⇒ 20 ⟨ (x+5)/4 + (x-3)/5  = 20 ⟨ (5x-8)/2 
⇒ 5(x+5)+4(x-3) = 10(5x-8)
⇒ 5x+25+4x-12 = 50x-80
⇒ 5x+4x-50x = -80-25+12
⇒ -41x = -93
⇒ x = -93/(-41)
Hence,
    x = 93/41

EXAMPLE 3. Find the value of x for the expression :-
                    (5x+7)/(2x+4)  =  (15x+2)/(6x+5) 
SOLUTION :
        (5x+7)/(2x+4)  =  (15x+2)/(6x+5)
    By cross multiply we get result-
        ⇒ (5x+7) (6x+5) = (2x+4) (15x+2)
        ⇒ 5x(6x+5)+7(6x+5) = 2x(15x+2)+4(15x+2)
        ⇒ 30x2+25x+42x+35 = 30x2+4x+60x+8
        ⇒ 30x2-30x2+25x+42x-4x-60x = 8-35
        ⇒ 3x = -27
        ⇒ x = -27/3 = -9
Hence,
        x = -9


(C). LINEAR EQUATIONS IN TWO VARIABLES




        1. STANDARD FORM OF LINEAR EQUATIONS IN TWO VARIABLE IS :-

            Y=mx + b
            Where, x,y = variables ,
                        m = Slop or gradient,
                        b = y- intercept

    We can represent linear equations in two variables on straight line in graph. Above equation also called "Slop Intercept equation of a straight line".
For Example, if  y=3x+5
                      then, Slop m=3 , Intercept b=5

        2. POINT-SLOP FORM OF EQUATION :-

            y-y1=m(x-x1)
    EXAMPLE :
              Y-5=2/3(X-2)
        Here,
              y1=5 , x1=2 , m=2/3

        2. GENERAL FORM OF STRAIGHT LINE:-

            Ax+By=C , or
              Ax+By+c=0
    Where,  A,B ≠ 0
        x,y are variables,
        A,B are coefficients, and
        c is constant
    Here, A,B,C are real numbers.

EXAMPLE :
        2x+3y-5=0
Where, A=2, B=3, C=-5

NOTE THAT :
1. These are not linear equations :-

        y2-5=0 ,
        y2/3=15 ,
        5√x-y=8
        
In linear equation exponents such as x2, x3 etc. and square roots or cube roots not allowed.

2. These are not linear equations :-

       y=3x+5 ,
        y+2=3(x+5) ,
        y-3x+5=0 ,
        3x=6 ,
        x/3 =5

        












          


          











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