Class 10th Mathematics Chapter 2 Polynomials

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Class 10 -> Mathematics -> Chapter 2: Polynomials

I. Chapter Summary:

Polynomials are expressions formed by adding terms that include variables raised to whole number powers. This chapter deals with the fundamental concepts of polynomials, including the degree, coefficients, and types of polynomials. It explores the relationship between the roots and coefficients of a polynomial, and introduces the factorization process and division of polynomials.

II. Key Concepts Covered:

  1. Polynomial Definition:
    • A polynomial is an algebraic expression consisting of variables raised to non-negative integer powers, and coefficients.
    • Standard form: $a_n x^n + a_{n-1} x^{n-1} + cdots + a_1 x + a_0$​, where $a_n, a_{n-1}, ldots, a_0$​ are constants and n is a non-negative integer.
  2. Degree of a Polynomial:
    • The degree is the highest power of the variable in a polynomial.
    • Example: The degree of $3x^2 + 5x + 2$ is 2.
  3. Types of Polynomials:
    • Monomial: A polynomial with only one term (e.g., $2x^3$).
    • Binomial: A polynomial with two terms (e.g., $3x^2 – 5x$).
    • Trinomial: A polynomial with three terms (e.g., $x^2 + 5x + 6$).
  4. Zero of a Polynomial:
    • A value of x that makes the polynomial equal to zero. These are also known as the roots of the polynomial.
  5. Remainder Theorem:
    • If a polynomial f(x) is divided by (x−a), the remainder is f(a).
  6. Factor Theorem:
    • A polynomial f(x) has a factor (x−a) if and only if f(a) = 0.
  7. Factorization of Polynomials:
    • The process of writing a polynomial as a product of its factors.
    • Example: $x^2 – 5x + 6 = (x – 2)(x – 3)$.

III. Important Questions:

(A) Multiple Choice Questions (MCQs) (1 Mark):

  1. What is the degree of the polynomial $4x^3 + 5x^2 – 3x + 7$?
    • (a) 3
    • (b) 2
    • (c) 1
    • (d) 0
    • Answer: (a) 3
  2. Which of the following is a factor of $x^2 – 3x + 2$?
    • (a) (x – 2)
    • (b) (x+2)
    • (c) (x−1)
    • (d) None
    • Answer: (a) (x−2)
  3. If x=2 is a root of the polynomial $x^2 – 4x + 3$ what is the value of f(2)?
    • (a) 0
    • (b) 3
    • (c) -3
    • (d) 2
    • Answer: (a) 0
  4. Which of the following is a polynomial?
    • (a) $frac{1}{x} + 2x$
    • (b) $2x + 3x^2 – x^3$
    • (c) $4x^{-2} + x$
    • (d) $2x + frac{3}{x} + x^3$​
    • Answer: (b) $2x + 3x^2 – x^3$

(B) Short Answer Questions (2/3 Marks):

  1. Find the zeroes of the polynomial $x^2 – 5x + 6$.
    • Answer: The zeroes are $x = 2 quad text{and} quad x = 3$.
  2. Using the Factor Theorem, find a factor of the polynomial $x^2 + 3x – 10 quad text{if} quad x – 2$ is a factor.
    • Answer: Factorization gives (x – 2)(x + 5).
  3. Use the Remainder Theorem to find the remainder when $frac{x^3 – 4x + 1}{x – 1}$.
    • Answer: The remainder is $f(1) = 1^3 – 4(1) + 1 = -2$.
  4. Factorize the polynomial $x^2 + 5x + 6$.
    • Answer: $x^2 + 5x + 6 = (x + 2)(x + 3)$.

(C) Long Answer Questions (5 Marks):

  1. Factorize the polynomial $x^3 – 3x^2 – 4x + 12$ by grouping.
    • Answer: $x^3 – 3x^2 – 4x + 12 = (x^2(x – 3) – 4(x – 3)) = (x – 3)(x^2 – 4) = (x – 3)(x – 2)(x + 2)$.
  2. Find the value of k for which $x^2 – 6x + k$ has a factor $x – 2$.
    • Answer: Using the Factor Theorem, $f(2) = 0$, gives $k = 4$.
  3. Divide $frac{2x^3 + 3x^2 – 5x + 6}{x + 2}$ using the long division method.
    • Answer: The quotient is $2x^2 – x – 3$, and the remainder is 0.
  4. Explain the Factor Theorem and solve the problem: If x+1x + 1x+1 is a factor of $x^3 – 2x^2 – x + 2$, find the other factors.
    • Answer: By Factor Theorem, divide the polynomial by $x + 1$ and find $(x – 2)(x^2 + x – 1)(x – 2)(x^2 + x – 1)$.

(D) HOTS (Higher Order Thinking Skills) Questions:

  1. Prove that the sum of the roots of a quadratic polynomial is equal to the negative of the coefficient of x, and the product of the roots is equal to the constant term.
    • Answer: Use the relation $ax^2 + bx + c = 0$, where the sum and product of roots are derived using Vieta’s formulas.
  2. A polynomial has roots −2 and 5. Construct a polynomial from its roots and find its degree.
    • Answer: The polynomial is $(x + 2)(x – 5) = x^2 – 3x – 10$, and the degree is 2.

IV. Key Formulas/Concepts:

  1. General Form of Polynomial: $a_n x^n + a_{n-1} x^{n-1} + cdots + a_1 x + a_0$​
  2. Remainder Theorem: If f(x) is divided by $(x – a)$, the remainder is f(a).
  3. Factor Theorem: $(x – a)$ is a factor of $f(x)$ if $f(a) = 0$.
  4. Sum of Roots (Quadratic): $text{Sum} = -frac{b}{a}$​
  5. Product of Roots (Quadratic): $text{Product} = frac{c}{a}$​

V. Deleted Portions (CBSE 2025-2026 as per rationalization of NCERT books from ncert.nic.in):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025-2026):

Unit/Chapter Estimated Marks Type of Questions Typically Asked
Polynomials 10-12 Marks MCQs (1 mark), Short Answer (2/3 marks), Long Answer (5 marks)

VII. Previous Year Questions (PYQs):

1 Mark Questions:

  • 2019: What is the degree of the polynomial $x^3 + 2x^2 – 5x + 6$
  • 2020: Which of the following is a factor of $x^2 – 6x + 9$?

2/3 Mark Questions:

  • 2018: Find the zeroes of the polynomial $x^2 – 4x + 3$.
  • 2021: Use the Factor Theorem to factorize $x^2 – 5x + 6$

5 Mark Questions:

  • 2019: Divide $2x^3 + 3x^2 – 5x + 6$ by $x + 1$ and find the quotient and remainder.
  • 2020: Factorize $x^3 – 3x^2 – 4x + 12$ using grouping.

VIII. Real-World Application Examples to Connect with Topics:

  1. Engineering: Polynomials are used in solving real-life problems involving optimization, such as maximizing or minimizing areas and volumes.
  2. Finance: Polynomials are applied in calculating compound interest and determining profit margins over time.

IX. Student Tips & Strategies for Success (Class-Specific):

  1. Time Management: Dedicate specific hours for studying polynomials and solve practice problems consistently.
  2. Exam Preparation: Focus on understanding the concept rather than memorizing formulas.
  3. Stress Management: Take breaks between study sessions and practice deep breathing exercises to stay calm during exams.

X. Career Guidance & Exploration (Class-Specific):

For Class 10th, explore options in Science (PCM), Commerce, and Arts. Mathematics is crucial for students opting for careers in engineering, economics, and data science. Entrance exams like JEE (for engineering), NEET (for medical), and various state-level exams are available for specific career paths.

XI. Important Notes:

  • Regular revision is key to mastering polynomials.
  • Focus on conceptual understanding to solve more complex problems.
  • Keep practicing previous year questions for better time management during exams.

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