নামটা শুনেই পিছিয়ে পড়ো না প্লিজ। বিশ্বাস করো , ব্যাপারটা সুন্দর। এডমিশনে আসা কঠিন ইন্টিগ্রেশনগুলোর একটা vital part হচ্ছে এই স্টেপটুকু। যদি আয়ত্তে নিয়ে আসতে পারো,লাভ ভিন্ন ক্ষতি হওয়ার কোনো কারণই দেখিনা। তো কথা না বাড়িয়ে কাজে চলে যাই!
আচ্ছা , তোমরা যখন দু’টো ভগ্নাংশকে যোগ করতে যাও,তখন কীভাবে করো? ভগ্নাংশগুলোর হরগুলোর ল.সা.গু নাও। তারপর যেটা করো,সেই ল.সা.গু কে ভগ্নাংশ দু’টোর হর দিয়ে ভাগ করে,ভাগফলটাকে লবগুলোর সাথে গুণ দিয়ে যোগ দিয়ে বসিয়ে দাও! মাথা ঘুরে গেল? দেখে আসি একটা উদাহরণঃ
দু’টো ভগ্নাংশকে যোগ করা হলো এখানে। এখন বিপত্তিটা বাঁধে কখন জানো? যখন আমরা এই যোগফল থেকে ভগ্নাংশ দু’টোকে আলাদা করতে যাবো। উপরের এই উদাহরণ থেকে আলাদাকরণ প্রক্রিয়াটা হয়তো অনেক সহজ! কারণ লব এর অংশটুকুকে উৎপাদকে ভেঙে ফেললেই কাজ একদম সহজ হয়ে যায়। কিন্তু কিছু কিছু যোগফল থেকে ভগ্নাংশগুলোকে টেনে হিচড়ে বের করাটা বেশ দুঃসাধ্য!
যেমনঃ
এই ভগ্নাংশটাকে উপরের নিয়মে Split করতে গেলেই বিপাকে পড়ে যাবে। Split এমনভাবেই করতে হবে,যেন প্রতিটা ভগ্নাংশের হরের অংশে (x-3) এবং (2x+1) এর উভয়েই না থাকে!
আলোচনার শুরুতে দু ধরণের ভগ্নাংশের ব্যাপারে একটু বলে নিই।
১) Proper Fraction
২) Improper Fraction
Proper Fraction হচ্ছে সেসব ভগ্নাংশ , যাদের লবের চাইতে হরে বিদ্যমান বহুপদীর মাত্রা বড়। যেমনঃ
আর Improper Fraction হচ্ছে সেসব ভগ্নাংশ , যাদের লবের চাইতে হরে বিদ্যমান বহুপদীর মাত্রা ছোট বা সমান। যেমনঃ
আজকের পর্বে আমরা দেখবো,একটি ভগ্নাংশ যদি Proper হয়,তাহলে তাকে কীভাবে কিছু ভগ্নাংশের সমষ্টি হিসেবে আলাদা করতে হয়।
Proper Fraction কে আলাদা করতে গেলে কয়েকটা বিষয় মাথায় রাখতে হয়। যেমনঃ
১) যখন হরে একই উৎপাদক একবারের চেয়ে বেশি থাকেনা।
২) যখন হরে একই উৎপাদক একবারের চেয়ে বেশি থাকে।
৩) যখন হরে উৎপাদকগুলোর মাঝে কেউ এমন থাকে,যে কিনা নিজেই বহুপদী!
১) যখন হরে একই উৎপাদক একবারের বেশি থাকেনা:
আমাদের উদাহরণের ভগ্নাংশটিকেই আবার টেনে নিয়ে আসি। ভগ্নাংশটার হরটার দু’টো কিউট কিউট উৎপাদক ছিলো,মনে আছে? যেহেতু এর উৎপাদক সংখ্যা দুই(২) , তাই একে আমরা দু’টো ভগ্নাংশের যোগফল হিসেবে ঠিক এভাবে উপস্থাপন করতে পারিঃ
——-(১)
খেয়াল রাখতে হবে,এখানে A আর B এর মান হবে ধ্রুবক কিছু। এর মাঝে চলককে নিয়ে আসা যাবেনা। যদি চলে আসে,তাহলে ওই ভগ্নাংশটুকুকে আবার Split করতে হবে।
শুরুতেই যেটা করবো , (১) এর বাম পক্ষের হর দিয়ে,পুরো ভগ্নাংশকে গুণ দিয়ে দিবো! এইযে নিচের মতো করেঃ
একটু কাটাকাটি করে আরেকটু ছোট করে ফেলার পর চেহারা যা দাঁড়ায়ঃ
——(2)
একটু যদি যোগ বিয়োগ গুণ করে ফেলি,তাহলে নিচের চেহারার দিকে এগোতে থাকবো!
উপরের সমীকরণের উভয়পক্ষ সত্য হওয়ার জন্যে 2A+B = 3 এবং A-3B = 5 হওয়া বাধ্যতামূলক! এই দু’টো সমীকরণ থেকে A এবং B এর মান বের করে নিলে যা দাড়ায়ঃ
A = 2 এবং B = -1
A এবং B এর মান দু’টো (১) এ বসিয়ে পাইঃ
হয়ে গেলো আংশিক ভগ্নাংশে প্রকাশ! খেয়াল করে দেখেছো,পুরো প্রক্রিয়াতে আমি একবারো ” আংশিক ভগ্নাংশ ” নামটি নিই নাই। কারণ অনেকে এই নামটা শুনলেই সুন্দর মতো স্ক্রল করে চলে যেতে। এতোই অপছন্দ অনেকের কাছে এই টপিক! :p
A এবং B এর মান অন্যভাবেও বের করা যেত। আমরা চাইলে (2) এ x=3 এবং x=(-1/2) বসিয়ে A এবং B এর মান বের করে নিতে পারতাম। একই কথা।
অনুশীলনের জন্যেঃ
২) যখন হরে একই উৎপাদক একবারের চেয়ে বেশি থাকে:
এবার ধরো একটা ভগ্নাংশ এমন,যার হরে একই উৎপাদক একের অধিকবার আছে। যেমনঃ
তখন যেভাবে যা ধরতে এবং করতে হবে তা হচ্ছেঃ
————–(1)
(x-1) যেহেতু দু’বার আছে,তাই একবার,দু’বার এভাবে মোট দুইটা আলাদা ভগ্নাংশে আমাদের ভাগ করে নিতে হবে। যদি তিনবার থাকতো,একইভাবে আমাদের একবার,দু’বার,তিনবার এভাবে ভাগ করে নিলেই হয়ে যেত।
আগের মতোই এবার (১) এর উভয়পক্ষকে (x-1)^2 (x+2) দিয়ে গুণ দিয়ে দিই!
———-(2)
সমীকরণ (২) এ x=1,-2 বসিয়ে কিছু মান বের করে ফেলা যাক!
বিপত্তি বেঁধে গেল মনে হচ্ছে? A এর মান কীভাবে বের করবে? উত্তরটা হচ্ছেঃ সহগ সমীকৃত করে!
(২) নং থেকে সহগ সমীকৃত করে পাইঃ
উপরের সমীকরণে ভালো করে তাকাও! দরকার শুধু A এর মান। তো x^2 এর সহগ হচ্ছে (A+C) , কিন্তু সমীকরণের বাম দিকে x^2 পদটাই অনুপস্থিত! তার মানে , এর সহগ হচ্ছে শূণ্য!
ধরাই যায়ঃ
A+C = 0
A = -C = 5/9
হয়ে গেল! A,B,C তিনজনের মানই জানা হয়ে গেল।
অনুশীলনের জন্যঃ
৩) যখন হরে উৎপাদকগুলোর মাঝে কেউ এমন থাকে,যে কিনা নিজেই বহুপদী:
এখন একটা ভগ্নাংশ হতে পারে এমনঃ
এর ক্ষেত্রে সমীকরণের ফরমেশন টা একটু আলাদা হবে,বাকি ক্যালকুলেশন সব আগের মতইঃ
——(1)
এইযে Ax+B হলো,এইটা সেই পার্টটুকুতেই হবে যার হর-এ তুমি বহুপদীটাকে বসাচ্ছো,BE CAREFUL!
এই সমস্যাটা নিজেরাই সমাধান করো! যদি দেখো A,B,C এর মান যথাক্রমে (-10/7) , (5/7) , (10/7) হয়েছে,তাহলে তোমার সমাধান সঠিক! মান বসিয়ে পেয়ে যাবে আংশিক ভগ্নাংশগুলো।
অনুশীলনের জন্যেঃ
অনুশীলনের সমস্যাগুলো সমাধান করে ফেলো। আগামী পর্বে Improper Fraction কে Decompose করা শিখে নিবো,সাথে জানিয়ে দিবো অনুশীলনের উত্তরগুলো! সেই পর্যন্ত সবাই ভালো থেকো!
কৃতজ্ঞতা স্বীকারঃ
1) mathcentre for providing such wonderful and vivid examples.
Sadman Sakib
Latest posts by Sadman Sakib (see all)
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স্বল্প সময়ে যেকোনোটার অংশ বের করার নিয়মটা বললে ভালো হয়।
Tnx
ভাল
Wow…… Thanks a million 🙂
khubbb valo hoyeche… awesome
kintu ekta qsn… jodi hor (x^2)/(x-1) hoy tokhn kivabe dhorbo????
ধন্যবাদ
পরের পর্বটা পাবো কিভা??
Please type a full example we can’t understand your problem
(x^2-1+1)÷(x-1)
=(x^2-1)÷(x-1)+{1÷(x-1)}
=(x^2-1^2)÷(x-1)+{1÷(x-1)}
={(x+1)(x-1)÷(x-1)}+{1÷(x-1)}
=(x+1)+{1÷(x-1)} (Ans.)
khub valo
vai … ssc higher math er (8.1-8.3) ektu bole dile valo hoyton..plz vai bolen
অনেক বেশি সহজ ও ভালো লাগছে।ধন্যবাদ সহজশিক্ষা প্রতিষ্ঠানকে।
সহগ সমীকৃত করণটা বুঝতে পারছিলাম না। এখন সলভড। ধন্যবাদ অনেক?
j tar deklam setai bujlam na??
অনেক ভালো। আমার সহগ সমীকৃত করতে একটু সমস্যা হতো।কিন্তু এখন ইনশাআল্লাহ আমি করতে পারবো।
অনেক সহজ হয়ে গেছে এখন।
ধন্যবাদ
কিন্তু পরের পর্বটা পাবো কিভা???
পরের পর্বটা পাবো কিভা??
আচ্ছা সহগ সমীকৃত সংজ্ঞা কী?
দয়া করে বলেন সহগ সমীকৃত কী?
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