Polynomials — الرياضيات – الصف 10 (CBSE) بشرح مبسّط

ما ستتعلّمه

1. 1

   الدرس 1: From Degree to Graphical Zeroes

   6 دقائق
   • Recall the Degree of a Polynomial
   • Recognise Linear Polynomials
   • Quadratic and Cubic: Names and General Forms
   • What Does the Value of a Polynomial Mean?
   • A Zero Makes the Polynomial's Value Zero
   • The Zero of a Linear Polynomial
   • A Linear Zero Is an x-Coordinate
   • From a Quadratic Value Table to Its Zeroes
2. 2

   الدرس 2: Counting Zeroes and Discovering Relationships

   5 دقائق
   • Two Distinct Zeroes or One Zero
   • No Zeroes and the Meaning of At Most
   • A Cubic with Three Zeroes
   • A Cubic Need Not Have Three Distinct Zeroes
   • Example 1: Count the Intersections
   • Factorise the First Quadratic
   • Compare the Zeroes with the Coefficients
3. 3

   الدرس 3: Zeroes and Coefficients: Quadratic Applications

   5 دقائق
   • A Second Quadratic Confirms the Pattern
   • Why the Quadratic Relationships Hold
   • Example 2: Find and Verify the Zeroes
   • Example 3: Zeroes Involving Square Roots
   • Example 4: Work Backwards from Sum and Product
   • A Cubic: Sum and Product of Three Zeroesغير مُقيَّم
   • The Third Relationship: Products Two at a Timeغير مُقيَّم
4. 4

   الدرس 4: Cubic Relationships, Enrichment and Practice

   5 دقائق
   • Organise the Three Cubic Relationshipsغير مُقيَّم
   • Optional Example 5: Verify the First Two Zeroesغير مُقيَّم
   • Optional Example 5: Verify the Fractional Zeroغير مُقيَّم
   • Optional Example 5: Verify the Sum Relationshipغير مُقيَّم
   • Optional Example 5: Verify Pair Products and the Productغير مُقيَّم
   • Exercise 2.2: Two Directions of Practice
   • Chapter Recap: Read, Relate and Practise

المفاهيم الأساسية

Linear classification

A polynomial of degree one is called a linear polynomial.

Quadratic and cubic forms

A polynomial of degree two is called a quadratic polynomial. The name comes from quadrate, meaning square.

Definition of a zero

A zero is an input number that makes the output zero. The input itself need not be zero.

Two quadratic intersection cases

Two separate markers give two distinct input values. One coincident marker gives one input value.

Quadratic upper bound

A quadratic can have two distinct zeroes, two equal zeroes described as one zero, or no zero.

Degree and maximum zero count

Degree provides a maximum, not an exact count.

تركيز الامتحان

Board exam

CBSE assesses this chapter in the 2026-27 year-end board examination.

Syllabus

POLYNOMIALS: 1. Zeros of a polynomial. 2. Relationship between zeros and coefficients of quadratic polynomials. Learning outcome: find the zeros of polynomial graphically and algebraically and verifying the relation between zeros and coefficients of quadratic polynomials.

Not assessed

These parts go beyond the 2026-27 syllabus and are taught for understanding only: A Cubic: Sum and Product of Three Zeroes, The Third Relationship: Products Two at a Time, Organise the Three Cubic Relationships, Optional Example 5: Verify the First Two Zeroes, Optional Example 5: Verify the Fractional Zero, Optional Example 5: Verify the Sum Relationship and Optional Example 5: Verify Pair Products and the Product.

Exam note

The relationship between zeroes and coefficients is in the syllabus for QUADRATIC polynomials only; the textbook's cubic relationships (and the cubic worked example) are outside the 2026-27 syllabus.

Paper pattern

Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.

Question style

About 50% of board questions are competency-focused.

Practice in the app

67 board-pattern questions with marking schemes: 21 multiple choice (1 mark), 10 assertion–reason (1 mark), 15 very short answer (2 marks), 10 short answer (3 marks), 4 long answer (5 marks) and 7 case-based (4 marks).

أسئلة نموذجية مع نقاط التصحيح

أسئلة بنمط امتحان المجلس من مجموعة تدريبات AI Lessons لهذا الفصل. توضّح نقاط التصحيح ما يبحث عنه المصحّح؛ والإجابات النموذجية الكاملة في التطبيق.

1. Very short answerدرجتان

   For p(x) = x² - 3x - 4, a student writes p(-1) = -1 + 3 - 4 = -2. Identify the error and evaluate p(-1) correctly.

   نقاط التصحيح

   • Identify the incorrect squaring of -1 and state that (-1)² = 1.1
   • Correctly substitute and obtain p(-1) = 0.1

2. Short answer3 درجات

   Using 3x² + 5x - 2 = (3x - 1)(x + 2), a student reports the zeroes as 1/3 and 2. Correct the error and verify both relationships between the zeroes and coefficients.

   نقاط التصحيح

   • Sets 3x - 1 = 0 and x + 2 = 0.0.5
   • Obtains the correct zeroes 1/3 and -2.0.5
   • Calculates the sum as -5/3 and compares it with -b/a = -5/3.1
   • Calculates the product as -2/3 and compares it with c/a = -2/3.1

3. Long answer5 درجات

   For p(x) = 2x² - 8x + 6, use a valid middle-term split to find the zeroes and verify their sum and product against the coefficients. Without plotting the graph, state all its x-axis intersection points and justify why there cannot be another such point.

   نقاط التصحيح

   • Chooses -6x and -2x and checks their sum -8x and product 12x².0.5
   • Completes the factorisation through grouping to obtain 2(x - 1)(x - 3).1
   • Obtains the zeroes 1 and 3.0.5
   • Verifies the sum using 1 + 3 = 4 = -(-8)/2.1
   • Verifies the product using 1 × 3 = 3 = 6/2.1
   • States both x-axis intersection points: (1, 0) and (3, 0).0.5
   • Justifies that no additional x-axis intersection is possible because a quadratic has at most two distinct real zeroes.0.5

الأسئلة الشائعة

فصول أخرى في الرياضيات – الصف 10

• الفصل 3Pair of Linear Equations in Two Variablesالدروس: 4 · الدقائق: 24→
• الفصل 4Quadratic Equationsالدروس: 4 · الدقائق: 22→
• الفصل 7Coordinate Geometryالدروس: 3 · الدقائق: 21→
• الفصل 8Introduction to Trigonometryالدروس: 4 · الدقائق: 26→

كل أدلة الرياضيات – الصف 10 (CBSE) →