Class 10 Notes

Class 10 Mathematics Notes Chapter 2 (Polynomials) – Mathematics Book

Philoid

Philoid

— 5 min read

Mathematics
Alright class, let's get straight into Chapter 2: Polynomials. This is a fundamental chapter, and understanding it well is crucial not just for your Class 10 exams but also forms the base for many concepts you'll encounter in higher mathematics and competitive exams. Pay close attention.

Chapter 2: Polynomials - Detailed Notes

1. Introduction to Polynomials

  • Definition: An algebraic expression p(x) of the form:
    p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + ... + a_1 x + a_0
    is called a polynomial in variable x, provided:
    • a_0, a_1, a_2, ..., a_n are real numbers (called coefficients).
    • a_n ≠ 0 (the coefficient of the highest power term is non-zero).
    • The exponents of the variable x (i.e., n, n-1, ..., 1, 0) are non-negative integers.
  • Examples: 2x + 3, 4y² - 3y + 7, 5t³ - √2 t² + 1/3
  • Non-Examples: 1/x (since x⁻¹, exponent is negative), √x + 3 (since x^(1/2), exponent is not an integer), x² + 1/(x-1) (not in the standard form, involves division by a variable expression).
  • Degree of a Polynomial: The highest power of the variable in a polynomial is called its degree.
    • Example: In p(x) = 5x³ - 2x² + x - 8, the highest power of x is 3. So, the degree is 3.
    • The degree of a non-zero constant polynomial (e.g., p(x) = 7) is 0.
    • The degree of the zero polynomial (p(x) = 0) is not defined.

2. Types of Polynomials (Based on Degree)

  • Linear Polynomial: A polynomial of degree 1. General form: ax + b, where a ≠ 0.
    • Example: 3x - 5, √2 y + 1
  • Quadratic Polynomial: A polynomial of degree 2. General form: ax² + bx + c, where a ≠ 0.
    • Example: x² - 5x + 6, 3y² - 8
  • Cubic Polynomial: A polynomial of degree 3. General form: ax³ + bx² + cx + d, where a ≠ 0.
    • Example: 2x³ - x² + 4x - 1, t³ + 7t

(Note: Polynomials can also be classified by the number of terms: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms).)

3. Value of a Polynomial

  • If p(x) is a polynomial in x, and k is any real number, then the value obtained by replacing x by k in p(x) is called the value of p(x) at x = k, denoted by p(k).
  • Example: If p(x) = x² - 3x + 2, find p(2).
    p(2) = (2)² - 3(2) + 2 = 4 - 6 + 2 = 0.

4. Zeroes of a Polynomial

  • A real number k is called a zero (or root) of the polynomial p(x) if p(k) = 0.
  • Finding the zeroes of a polynomial p(x) means solving the equation p(x) = 0.
  • Example: For p(x) = x² - 3x + 2, we found p(2) = 0. So, x = 2 is a zero of p(x). Also, p(1) = (1)² - 3(1) + 2 = 1 - 3 + 2 = 0. So, x = 1 is also a zero of p(x).

5. Geometrical Meaning of the Zeroes of a Polynomial

  • The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
  • Linear Polynomial (ax + b): The graph is a straight line. It intersects the x-axis at exactly one point, (-b/a, 0). So, a linear polynomial has exactly one zero (x = -b/a).
  • Quadratic Polynomial (ax² + bx + c): The graph is a parabola.
    • If a > 0, the parabola opens upwards (∪).
    • If a < 0, the parabola opens downwards (∩).
    • The parabola can intersect the x-axis at:
      • Two distinct points: Two distinct zeroes.
      • Exactly one point (touches the x-axis): Two equal zeroes (one distinct zero).
      • No points: No real zeroes.
    • Therefore, a quadratic polynomial has at most two zeroes.
  • Cubic Polynomial (ax³ + bx² + cx + d): The graph is a curve that can intersect the x-axis at:
    • One point
    • Two points
    • Three points
    • Therefore, a cubic polynomial has at most three zeroes.
  • In general: A polynomial of degree n has at most n real zeroes.

6. Relationship Between Zeroes and Coefficients of a Polynomial

  • Linear Polynomial (p(x) = ax + b):
    • Zero is k = -b/a = -(Constant term) / (Coefficient of x)
  • Quadratic Polynomial (p(x) = ax² + bx + c):
    • Let the zeroes be α and β.
    • Sum of zeroes: α + β = -b/a = -(Coefficient of x) / (Coefficient of x²)
    • Product of zeroes: αβ = c/a = (Constant term) / (Coefficient of x²)
    • Forming a Quadratic Polynomial: If the sum (S = α + β) and product (P = αβ) of zeroes are given, the quadratic polynomial can be formed as:
      p(x) = k [x² - (α + β)x + αβ] = k [x² - Sx + P], where k is any non-zero real number (usually taken as 1 if not specified).
  • Cubic Polynomial (p(x) = ax³ + bx² + cx + d):
    • Let the zeroes be α, β, and γ.
    • Sum of zeroes: α + β + γ = -b/a = -(Coefficient of x²) / (Coefficient of x³)
    • Sum of the product of zeroes taken two at a time: αβ + βγ + γα = c/a = (Coefficient of x) / (Coefficient of x³)
    • Product of zeroes: αβγ = -d/a = -(Constant term) / (Coefficient of x³)
      (Note: The relationship for cubic polynomials might be beyond the core scope for some exams based on recent syllabus rationalization, but it's good to be aware of.)

7. Division Algorithm for Polynomials

  • If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) (quotient) and r(x) (remainder) such that:
    p(x) = g(x) × q(x) + r(x)
    where either r(x) = 0 or degree(r(x)) < degree(g(x)).
  • This is analogous to Euclid's division lemma for integers (Dividend = Divisor × Quotient + Remainder).
  • Process: Use polynomial long division.
  • Application:
    • If the remainder r(x) = 0 when p(x) is divided by g(x), then g(x) is a factor of p(x).
    • If some zeroes of a polynomial are known, we can use the division algorithm to find the remaining zeroes. For example, if x = k is a zero of p(x), then (x - k) is a factor. We can divide p(x) by (x - k) to get a quotient polynomial of a lower degree, whose zeroes can then be found.

Key Takeaways for Exams:

  1. Be able to identify polynomials and their degrees.
  2. Understand the meaning of zeroes and how to find them graphically (x-intercepts).
  3. Memorize and apply the relationship between zeroes and coefficients, especially for quadratic polynomials.
  4. Be able to form a quadratic polynomial given the sum and product of its zeroes.
  5. Understand and apply the Division Algorithm, particularly to check for factors and find remaining zeroes.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 2 for your practice:

  1. Which of the following is a polynomial?
    (a) x² + 1/x²
    (b) √x - 5
    (c) x³ - 3x² + √2 x + 1
    (d) x + 1/(x+1)

  2. The degree of the polynomial p(x) = 7x³ - 5x⁵ + 3x + 9 is:
    (a) 1
    (b) 3
    (c) 5
    (d) 7

  3. If p(x) = x² - 2x - 3, then the value of p(-1) is:
    (a) -4
    (b) 0
    (c) -2
    (d) 6

  4. The zeroes of the polynomial p(x) = x² - 4x + 3 are:
    (a) 1, 3
    (b) -1, 3
    (c) 1, -3
    (d) -1, -3

  5. A quadratic polynomial whose sum and product of zeroes are 5 and 6 respectively is:
    (a) x² + 5x + 6
    (b) x² - 5x + 6
    (c) x² - 5x - 6
    (d) x² + 5x - 6

  6. If α and β are the zeroes of the polynomial 2x² + 5x - 3, then the value of α + β is:
    (a) 5/2
    (b) -3/2
    (c) 3/2
    (d) -5/2

  7. If α and β are the zeroes of the polynomial x² - 6x + k, and αβ = 8, then the value of k is:
    (a) -6
    (b) 6
    (c) 8
    (d) -8

  8. The graph of a polynomial y = p(x) intersects the x-axis at 3 distinct points. What can be the degree of the polynomial p(x)?
    (a) 2
    (b) At least 3
    (c) At most 2
    (d) Exactly 1

  9. On dividing x³ + 3x² + 3x + 1 by x + 1, the remainder is:
    (a) 1
    (b) 0
    (c) -1
    (d) 2

  10. If one zero of the quadratic polynomial kx² + 3x + k is 2, then the value of k is:
    (a) -6/5
    (b) 6/5
    (c) 5/6
    (d) -5/6

Answers to MCQs:

  1. (c)
  2. (c)
  3. (b)
  4. (a)
  5. (b)
  6. (d)
  7. (c)
  8. (b)
  9. (b)
  10. (a)

Study these notes thoroughly. Practice solving problems from the NCERT textbook and previous year question papers focusing on these concepts. Let me know if any part needs further clarification. Good luck!

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