How To Solve LCM in Polynomials

Duration: 13 min

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Module outline

  1. Child Development and Pedagogy: Concept of Pedagogy, बालक कैसे सोचते एवं सीखते हैं / How Children Think and Learn, विकास की अवधारणा / Concept of Development, विकास के मनोवैज्ञानिक सिद्धांत / Psychological Theories of Development, Personality Development, व्यक्तिगत भिन्नताएँ, व्यक्तित्व एवं जेंडर / Individual Differences, Personality and Gender, संवेग, अभिप्रेरणा, मानसिक स्वास्थ्य एवं निर्देशन / Emotion, Motivation, Mental Health and Guidance, Understanding Diverse Learners, समावेशी शिक्षा / Inclusive Education, शिक्षण-अधिगम प्रक्रिया / Teaching-Learning Process, शिक्षण विधियाँ एवं रणनीतियाँ / Teaching Methods and Strategies, Experiential Pedagogy, आकलन एवं मूल्यांकन / Assessment and Evaluation, शैक्षिक विचारक, उद्देश्य एवं राष्ट्रीय नीतियाँ / Educational Thinkers, Aims and National Policies, Curriculum Framework & Reforms, Miscellaneous Topics, Nature, Objectives & Characteristics of Teaching, Learners & Learning Process, Factors Affecting Teaching, कक्षा प्रबंधन और सम्प्रेषण / Classroom Management and Communication, शिक्षण संसाधन और शैक्षिक प्रौद्योगिकी / Teaching Resources and Educational Technology, पाठ्यचर्या और विद्यालयी गतिविधियाँ / Curriculum and School Activities, पाठ योजना और शिक्षण कौशल / Lesson Planning and Teaching Skills, Evaluation, Assessment & Measurement, Inclusive Education & Special Needs, Guidance, Counselling & Mental Health, संविधान, आयोग, अधिनियम एवं योजनाएँ / Constitution, Commissions, Acts and Schemes in Education, अधिगम एवं इसके सिद्धांत / Learning and Its Theories, बुद्धि एवं सृजनात्मकता / Intelligence and Creativity, शिक्षक पहचान और व्यावसायिक विकास / Teacher Identity and Professional Development, जीवन कौशल, मूल्य, शांति एवं पर्यावरण शिक्षा / Life Skills, Values, Peace and Environmental Education, पर्यावरण अध्ययन का शिक्षणशास्त्र / Pedagogy of Environmental Studies, विज्ञान शिक्षण शास्त्र / Science Pedagogy, विद्यालय नेतृत्व और प्रशासन / School Leadership and Administration, कंप्यूटर हार्डवेयर एवं परिधीय उपकरण / Computer Hardware and Peripherals
  2. Language I — Hindi: भाषा, लिपि और वर्ण-विचार, संधि, शब्द-स्रोत और शब्द-रचना, समास, शब्द-भेद: विकारी और अविकारी, लिंग, वचन, कारक, काल और वाच्य, वाक्य-विचार और शुद्ध लेखन, शब्द-भंडार, रस, छंद, अलंकार और शब्द-शक्ति, हिंदी साहित्य, अपठित बोध, हिंदी शिक्षणशास्त्र
  3. Language Pedagogy (Language I and II): Language Pedagogy (Hindi/English/Sanskrit/Urdu learning & teaching)
  4. Language II — English: Sounds, Spelling and Word Formation, Parts of Speech, Nouns and Pronouns, Articles, Determiners and Adjectives, Verbs and Modals, Tenses and Subject–Verb Agreement, Adverbs, Prepositions and Conjunctions, Sentence Structure, Voice and Narration, Vocabulary, Sentence Correction and Writing Skills, Poetry and English Literature, Reading Comprehension, Pedagogy of English Language
  5. Language II — Urdu: رسم الخط، املا اور تلفظ, ذخیرۂ الفاظ, قواعد, محاورے اور ضرب الامثال, شعر و سخن, اردو ادب کی تاریخ, تفہیمِ عبارت, اردو کی تدریس
  6. Language II — Sanskrit: वर्णविचारः एवं संज्ञाप्रकरणम्, सन्धिप्रकरणम्, शब्दरूपाणि, धातुरूपाणि, कारकाणि एवं विभक्तयः, उपसर्गाः एवं अव्ययानि, प्रत्ययप्रकरणम्, समासप्रकरणम्, वाच्यप्रकरणम्, संख्याज्ञानम्, शब्दज्ञानम् एवं अनुवादः, अपठितावबोधनम्, संस्कृतसाहित्यम् एवं काव्यशास्त्रम्, संस्कृतशिक्षणशास्त्रम्
  7. Mathematics: Mathematics Pedagogy, संख्या पद्धति एवं स्थानीय मान / Number System & Place Value, मूलभूत संक्रियाएँ, सरलीकरण एवं पैटर्न / Basic Operations, Simplification & Patterns, गुणनखंड, विभाज्यता, म.स.प. एवं ल.स.प. / Factors, Divisibility, HCF & LCM, भिन्न एवं दशमलव / Fractions & Decimals, वर्ग, घन, घातांक एवं करणी / Squares, Cubes, Exponents & Surds, मापन: लंबाई, भार, धारिता, समय एवं मुद्रा / Measurement: Length, Mass, Capacity, Time & Money, बीजगणित / Algebra, अनुपात, समानुपात, औसत एवं आयु / Ratio, Proportion, Average & Ages, प्रतिशत, लाभ-हानि एवं ब्याज / Percentage, Profit–Loss & Interest, समय-कार्य एवं चाल-दूरी / Time & Work and Speed–Distance, रेखाएँ, कोण, आकृतियाँ एवं निर्देशांक / Lines, Angles, Shapes & Coordinates, त्रिभुज, चतुर्भुज, वृत्त एवं त्रिकोणमिति / Triangles, Quadrilaterals, Circles & Trigonometry, क्षेत्रमिति: परिमाप, क्षेत्रफल एवं आयतन / Mensuration: Perimeter, Area & Volume, आँकड़ा प्रबंधन, सांख्यिकी एवं प्रायिकता / Data Handling, Statistics & Probability, गणित शिक्षणशास्त्र / Mathematics Pedagogy
  8. Science: Science Pedagogy, प्रौद्योगिकी, मापन एवं गति / Technology, Measurement and Motion, बल, दाब, कार्य एवं ऊर्जा / Force, Pressure, Work and Energy, ऊष्मा एवं ध्वनि / Heat and Sound, प्रकाश / Light, विद्युत एवं चुंबकत्व / Electricity and Magnetism, ब्रह्मांड एवं प्राकृतिक परिघटनाएँ / Universe and Natural Phenomena, पदार्थ एवं उसकी संरचना / Matter and Its Structure, रासायनिक अभिक्रियाएँ, अम्ल, क्षार एवं लवण / Chemical Reactions, Acids, Bases and Salts, धातु, अधातु एवं दैनिक उपयोग की सामग्री / Metals, Non-metals and Materials of Daily Use, सजीव जगत: वर्गीकरण, कोशिका एवं ऊतक / Living World: Classification, Cell and Tissues, पादपों में पोषण, परिवहन एवं जनन / Plants: Nutrition, Transport and Reproduction, मानव शरीर एवं अंग तंत्र / Human Body and Organ Systems, भोजन, पोषण एवं स्वास्थ्य / Food, Nutrition and Health, पर्यावरण एवं पारिस्थितिकी / Environment and Ecology
  9. Social Studies/Social Science: प्राचीन भारत का इतिहास / Ancient Indian History, मध्यकालीन भारत का इतिहास / Medieval Indian History, औपनिवेशिक भारत एवं समाज सुधार / Colonial India and Social Reform, भारतीय राष्ट्रीय आंदोलन एवं स्वतंत्र भारत / Indian National Movement and Independent India, विश्व इतिहास एवं अंतरराष्ट्रीय संबंध / World History and International Relations, पृथ्वी, ग्लोब एवं मानचित्र / Earth, Globe and Maps, भौतिक भूगोल / Physical Geography, प्राकृतिक वनस्पति, वन्य जीवन एवं विश्व के प्रदेश / Natural Vegetation, Wildlife and World Regions, भारत का भूगोल / Geography of India, संसाधन एवं विकास / Resources and Development, सामाजिक एवं राजनीतिक जीवन / Social and Political Life, भारतीय संविधान / Indian Constitution, सरकार: केंद्र, राज्य एवं स्थानीय / Government: Union, State and Local, Social Studies Pedagogy, उत्तर प्रदेश सामान्य ज्ञान / Uttar Pradesh GK, अर्थशास्त्र एवं भारतीय अर्थव्यवस्था / Economics and Indian Economy, सामाजिक विज्ञान शिक्षणशास्त्र / Social Science Pedagogy
  10. Mathematics & Science (Video Lessons & Notes): Number System, Divisibility Rules, HCF LCM, Powers and Exponents (Surds and Indices), Algebra, Sequence and Series (Series), Ratio and Proportion (Ratios), Percentage, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Pipe and Cistern, Mensuration and Geometry, Partnership, Statistics, Data Interpretation, Probability, Biology, Chemistry, Physics
  11. Social Studies (Video Lessons & Notes): Geography, History, Polity, Environment, Static General Knowledge, State Wise GK
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video provides a comprehensive lesson on finding the Least Common Multiple (LCM) and Highest Common Factor (HCF) of polynomials. The lecture begins by defining LCM as the smallest polynomial divisible by all given polynomials and demonstrates its calculation through factoring, using examples like P(x) = x² - 9 and Q(x) = x² - 3x. The video then introduces the relationship between HCF and LCM, presenting the formula HCF × LCM = Product of the given polynomials, and illustrates this with a worked example. A section on 'Shortcut Tricks' is provided, including rules for determining HCF based on degree, common factors, and division. The video concludes with a practice question asking for the HCF of x² - 9 and x² - 3x, reinforcing the concepts taught.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title slide, "What is LCM of Polynomials?". It defines LCM as the smallest polynomial that can be divided by all given polynomials. Two examples are shown. Example 1 factors P(x) = x² - 9 as (x-3)(x+3) and Q(x) = x² - 3x as x(x-3). The LCM is determined by taking the highest power of each unique factor, resulting in LCM = x(x-3)(x+3). Example 2 follows a similar process for P(x) = x² - 5x + 6 and Q(x) = x² - 3x, yielding LCM = x(x-2)(x-3).

  2. 2:00 – 5:00 02:00-05:00

    The video transitions to a comparison between the numeric and polynomial methods for finding HCF. It shows the Euclidean algorithm for numbers 48 and 18, where the HCF is found to be 6. This is contrasted with the polynomial method for P(x) = x³ - 3x² + 2x and Q(x) = x² - 2x + 1, where the HCF is found to be (x-1). The instructor then draws a diagram to illustrate the relationship between the factors of the polynomials, showing how the HCF is the common part and the LCM is the product of all unique factors.

  3. 5:00 – 10:00 05:00-10:00

    The instructor uses a diagram to explain the relationship between the factors of the polynomials. He writes the factors of P(x) = x² - 9 as (x-3) and (x+3), and the factors of Q(x) = x² - 3x as x and (x-3). He then shows that the HCF is the common factor (x-3), and the LCM is the product of all unique factors: x(x-3)(x+3). He also demonstrates the relationship between the HCF and LCM using the formula HCF × LCM = Product of the given polynomials, using the example of 20 and 30 to illustrate the concept.

  4. 10:00 – 12:50 10:00-12:50

    The video presents a slide titled "Relation Between HCF and LCM" with the formula: HCF × LCM = Product of the given polynomials. An example is worked out using P(x) = x² - 9 and Q(x) = x² - 3x, showing that HCF = (x-3) and LCM = x(x-3)(x+3). The instructor verifies the formula by multiplying HCF and LCM to get the product of the polynomials. The final section, "Shortcut Tricks," lists five rules: 1) HCF cannot have a degree greater than the smallest polynomial, 2) check for common factors like (x+a) or (x-b), 3) use the formula LCM = (Product / HCF), 4) if one polynomial divides the other, the smaller is the HCF and the larger is the LCM, and 5) always verify the answer using HCF × LCM = Product. The video ends with a practice question: "Find the HCF of x² - 9 and x² - 3x."

The video systematically teaches the concepts of LCM and HCF for polynomials. It starts with a clear definition and two worked examples to demonstrate the factoring method. It then draws a powerful analogy to the numeric method, using the Euclidean algorithm to show the parallel process. The core of the lesson is the relationship between HCF and LCM, which is presented as a fundamental formula. This is reinforced with a detailed example and a set of practical shortcut tricks that provide students with efficient problem-solving strategies. The progression from definition to method to formula to shortcuts creates a logical and comprehensive learning path.

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