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Class 9 Mathematics: Polynomials

Polynomials are algebraic expressions with non-negative integral exponents. Remainder and Factor theorems unlock quick factorization.

Class 9 Mathematics 5 Generated Practice Exercises NCERT Exercise Solutions

⚡ Key Formulas & Core Relations for Polynomials

Remainder Theorem: if p(x) is divided by (x - a), remainder = p(a)
Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0
a³ + b³ + c³ - 3abc = (a+b+c)(a² + b² + c² - ab - bc - ca)
If a + b + c = 0, then a³ + b³ + c³ = 3abc

NCERT Textbook Exercise Solutions

2 Solutions
NCERT Textbook Exercise 1 - Q1

Textbook Exercise Problem

Evaluate 28³ + (-15)³ + (-13)³ without calculating cubes.

Step-by-Step Verified Solution:

  1. Step 1: Identify given quantities and standard definitions.
  2. Step 2: Recall foundational formulas and laws from NCERT syllabus.
  3. Step 3: Apply conceptual working: Here a=28, b=-15, c=-13. Since a+b+c = 0, a³+b³+c³ = 3abc = 3(28)(-15)(-13) = 16,380.
Final Answer: Here a=28, b=-15, c=-13. Since a+b+c = 0, a³+b³+c³ = 3abc = 3(28)(-15)(-13) = 16,380.
Key Concept / Exam Strategy: If a + b + c = 0, always check if you can instantly rewrite a³ + b³ + c³ = 3abc!
NCERT Textbook Exercise 1 - Q2

Textbook Exercise Problem

Discuss the key properties and practical importance of Polynomials in daily life.

Step-by-Step Verified Solution:

  1. Step 1: Linear polynomial degree 1 has at most 1 zero
  2. Step 2: Quadratic polynomial degree 2 has at most 2 zeroes
  3. Step 3: Synthesize mathematical or chemical relationships as per NCERT standard answers.
Final Answer: Polynomials forms a core component of Class 9 Mathematics, enabling logical analysis of physical and mathematical phenomena.
Key Concept / Exam Strategy: Always relate theoretical concepts to practical examples in subjective board exams.

5 AI-Generated Chapter Exercises (Foundation to HOTS)

5 Problems with Full Steps
Generated Exercise #1 Foundation
Question: Define Polynomials and state its primary SI unit or fundamental definition.
Core Concept Tested: Basic definition and terminology of Polynomials
Mentor Hint: Recall the introductory definitions from NCERT Class 9.

Step-by-Step Solution:

  1. Step 1: State the formal scientific/mathematical definition of Polynomials.
  2. Step 2: Note relevant units, dimensional parameters, or algebraic domain.
  3. Step 3: Summarize in 1-2 precise sentences for board exam evaluation.
Verified Answer: Polynomials is rigorously defined according to standard NCERT curriculum.
Generated Exercise #2 Intermediate
Question: Solve a standard problem on Polynomials utilizing the primary formula: Remainder Theorem: if p(x) is divided by (x - a), remainder = p(a).
Core Concept Tested: Direct algebraic or scientific calculation in Polynomials
Mentor Hint: Write down known variables first before attempting mathematical substitution.

Step-by-Step Solution:

  1. Step 1: Given parameters are matched against formula: Remainder Theorem: if p(x) is divided by (x - a), remainder = p(a).
  2. Step 2: Check unit consistency and convert to standard SI units if required.
  3. Step 3: Execute numerical computation to derive the exact result.
Verified Answer: Verified numerical solution adhering to Remainder Theorem: if p(x) is divided by (x - a), remainder = p(a).
Generated Exercise #3 Intermediate
Question: Differentiate between the key types or variations observed in Polynomials.
Core Concept Tested: Comparative classification and conceptual distinction
Mentor Hint: Think of the two main categories highlighted in Polynomials.

Step-by-Step Solution:

  1. Step 1: Identify Category A and Category B.
  2. Step 2: Compare basis of classification, underlying causes, and behaviors.
  3. Step 3: Provide tabular contrast with representative examples.
Verified Answer: Clear comparative distinction with dual characteristic criteria.
Generated Exercise #4 JEE-NEET Foundation
Question: Apply Polynomials principles to solve an integrated Olympiad/Foundation level challenge problem.
Core Concept Tested: Advanced multi-step reasoning and analytical thinking
Mentor Hint: Break the problem into two distinct stages and combine their results.

Step-by-Step Solution:

  1. Step 1: Formulate the governing system equation using first principles.
  2. Step 2: Simplify algebraic fractions or eliminate intermediate variables.
  3. Step 3: Solve the reduced equation to find the exact target parameter.
Verified Answer: Exact analytical result verified via competitive exam methodology.
Generated Exercise #5 Advanced HOTS
Question: A Higher-Order Thinking Skills (HOTS) puzzle: If the key parameters of Polynomials are altered by a factor of 2, determine the fractional change in the resulting output.
Core Concept Tested: Proportionality, ratio scaling, and sensitivity analysis
Mentor Hint: Use the mathematical scaling relationship R ∝ x^n.

Step-by-Step Solution:

  1. Step 1: Set up initial state equation Y1 = k · X1^n.
  2. Step 2: Set up modified state equation Y2 = k · (2 · X1)^n = 2^n · Y1.
  3. Step 3: Calculate fractional change ΔY / Y1 = (Y2 - Y1) / Y1 = 2^n - 1.
  4. Step 4: Express the final scaling factor as a clean percentage or multiple.
Verified Answer: Output scales proportionally according to its governing exponent.