Class 9 Mathematics Notes Chapter 2 (Polynomials) – Mathematics Book

Alright class, let's dive straight into Chapter 2: Polynomials. This is a fundamental chapter, and understanding it well will be very helpful for your future studies and competitive exams. Pay close attention!

Chapter 2: Polynomials - Detailed Notes for Exam Preparation

1. Introduction to Polynomials

• Definition: A polynomial in one variable x is an algebraic expression of the form:
  p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
  where a_0, a_1, ..., a_n are constants (called coefficients) and n is a non-negative integer. The powers of the variable (x) must be whole numbers (0, 1, 2, 3,...).
• Key Conditions:
  • Coefficients (a_i) can be any real numbers.
  • Powers of the variable must be non-negative integers.
• Examples of Polynomials:
  • 2x + 5 (Polynomial in x)
  • 3y^2 - 7y + 4 (Polynomial in y)
  • 8 (Constant polynomial)
  • 5t^3 - t (Polynomial in t)
• Examples of Expressions that are NOT Polynomials:
  • x + 1/x (because 1/x = x^{-1}, power is negative)
  • √t + 5 (because √t = t^{1/2}, power is not an integer)
  • y^{2/3} + 3 (power is not an integer)

2. Terminology

• Terms: The parts of a polynomial separated by + or - signs are called terms. For example, in 3x^2 - 5x + 7, the terms are 3x^2, -5x, and 7.
• Coefficient: The numerical part of a term is its coefficient. In 3x^2 - 5x + 7, the coefficient of x^2 is 3, the coefficient of x is -5, and the constant term (coefficient of x^0) is 7.
• Constant Polynomial: A polynomial with only one term, which is a constant (like 5, -2, 0).
• Zero Polynomial: The constant polynomial 0 is called the zero polynomial. Its degree is not defined.

3. Degree of a Polynomial

• Definition: The highest power of the variable in a polynomial is called its degree.
• Examples:
  • p(x) = 7x^3 - 4x^2 + x - 2: Degree is 3.
  • q(y) = 5y^6 - 3y^4 + y: Degree is 6.
  • r(t) = 9t - 1: Degree is 1.
  • s(x) = 10: Degree is 0 (since 10 = 10x^0).
• Degree of Zero Polynomial: The degree of the zero polynomial (0) is not defined.

4. Types of Polynomials

• Based on Number of Terms:
  • Monomial: A polynomial with only one term (e.g., 5x^2, -3y, 7).
  • Binomial: A polynomial with exactly two terms (e.g., x + 1, 3t^2 - 5).
  • Trinomial: A polynomial with exactly three terms (e.g., 2x^2 + 5x - 1, y^4 - y + 6).
• Based on Degree:
  • Linear Polynomial: A polynomial of degree 1. Standard form: ax + b, where a ≠ 0 (e.g., 2x - 3, y + √2).
  • Quadratic Polynomial: A polynomial of degree 2. Standard form: ax^2 + bx + c, where a ≠ 0 (e.g., x^2 - 5x + 6, 3y^2 - 7).
  • Cubic Polynomial: A polynomial of degree 3. Standard form: ax^3 + bx^2 + cx + d, where a ≠ 0 (e.g., 4x^3 - x^2 + 2x - 9).
  • Constant Polynomial: A polynomial of degree 0 (e.g., 7, -1/2). (Excludes the zero polynomial).

5. Zeroes of a Polynomial

• Definition: A real number k is called a zero (or root) of a polynomial p(x) if p(k) = 0.
• Finding Zeroes: To find the zero(es) of a polynomial p(x), set p(x) = 0 and solve for x.
• Example: Find the zero of p(x) = 2x + 6.
  Set p(x) = 0 => 2x + 6 = 0 => 2x = -6 => x = -3.
  So, -3 is the zero of the polynomial 2x + 6.
• Important Points:
  • A non-zero constant polynomial has no zero.
  • The zero polynomial has every real number as its zero.
  • A linear polynomial (ax + b, a ≠ 0) has exactly one zero: x = -b/a.
  • A polynomial of degree n can have at most n zeroes.

6. Remainder Theorem

• Statement: Let p(x) be any polynomial of degree greater than or equal to 1, and let a be any real number. If p(x) is divided by the linear polynomial (x - a), then the remainder is p(a).
• Significance: It allows us to find the remainder without performing the long division.
• Example: Find the remainder when p(x) = x^3 - 2x^2 + x + 1 is divided by x - 1.
  Here, the divisor is x - 1, so a = 1.
  By Remainder Theorem, the remainder is p(1).
  p(1) = (1)^3 - 2(1)^2 + (1) + 1 = 1 - 2 + 1 + 1 = 1.
  So, the remainder is 1.
• Note: If dividing by (x + a), the remainder is p(-a). If dividing by (ax - b), the remainder is p(b/a).

7. Factor Theorem

• Statement: Let p(x) be a polynomial of degree n ≥ 1 and a be any real number.
  1. (x - a) is a factor of p(x) if p(a) = 0.
  2. p(a) = 0 if (x - a) is a factor of p(x).
• Connection: The Factor Theorem directly links the zeroes of a polynomial to its factors. If k is a zero of p(x), then (x - k) is a factor of p(x).
• Example: Check if (x + 2) is a factor of p(x) = x^3 + 3x^2 + 5x + 6.
  The zero of x + 2 is x = -2.
  We need to find p(-2).
  p(-2) = (-2)^3 + 3(-2)^2 + 5(-2) + 6 = -8 + 3(4) - 10 + 6 = -8 + 12 - 10 + 6 = 0.
  Since p(-2) = 0, by the Factor Theorem, (x + 2) is a factor of p(x).

8. Factorization of Polynomials

• Using Common Factors: Factor out the greatest common factor from all terms.
• Factorization by Grouping: Group terms strategically to find common factors.
• Splitting the Middle Term (for Quadratic Trinomials ax^2 + bx + c):
  1. Find two numbers, say p and q, such that p + q = b (the coefficient of x) and p * q = a * c (the product of the coefficient of x^2 and the constant term).
  2. Rewrite the middle term bx as px + qx.
  3. Factor by grouping the first two terms and the last two terms.
  • Example: Factorize x^2 + 5x + 6.
    Here a=1, b=5, c=6. We need p+q=5 and p*q = 1*6=6. The numbers are 2 and 3.
    x^2 + 5x + 6 = x^2 + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
• Using the Factor Theorem (especially for Cubic Polynomials):
  1. Find one zero a of p(x) by testing factors of the constant term (using integer root theorem, if applicable, or simple trial).
  2. If p(a) = 0, then (x - a) is a factor.
  3. Divide p(x) by (x - a) using long division to get a quadratic quotient q(x).
  4. Factorize the quadratic quotient q(x) using splitting the middle term or identities.
  5. The complete factorization is (x - a) * (factors of q(x)).
• Using Algebraic Identities: (See next section)

9. Algebraic Identities

These are crucial for both expansion and factorization. You MUST memorize them.

1. (a + b)^2 = a^2 + 2ab + b^2
2. (a - b)^2 = a^2 - 2ab + b^2
3. a^2 - b^2 = (a + b)(a - b)
4. (x + a)(x + b) = x^2 + (a + b)x + ab
5. (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
6. (a + b)^3 = a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3
7. (a - b)^3 = a^3 - b^3 - 3ab(a - b) = a^3 - 3a^2b + 3ab^2 - b^3
8. a^3 + b^3 = (a + b)(a^2 - ab + b^2)
9. a^3 - b^3 = (a - b)(a^2 + ab + b^2)
10. a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
    • Special Case: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.

Key Takeaways for Exams:

• Be absolutely clear on the definition of a polynomial and its degree.
• Master the Remainder and Factor Theorems – they are frequently tested.
• Practice factorization techniques, especially splitting the middle term and using identities.
• Memorize all the algebraic identities thoroughly.

Multiple Choice Questions (MCQs)

1. Which of the following expressions is a polynomial in one variable?
   (a) x + 1/x
   (b) √y + 3
   (c) 3z^2 - 5z + √7
   (d) t^2 + t^{3/2}

2. The degree of the polynomial p(x) = 5x^4 - 0x^5 + 3x + 7 is:
   (a) 5
   (b) 4
   (c) 1
   (d) 0

3. The zero of the linear polynomial f(x) = 3x + 5 is:
   (a) 5/3
   (b) -5/3
   (c) 3/5
   (d) -3/5

4. What is the remainder when x^3 - 1 is divided by x - 1?
   (a) -1
   (b) 0
   (c) 1
   (d) 2

5. If (x + 1) is a factor of the polynomial 2x^2 + kx, then the value of k is:
   (a) -2
   (b) -3
   (c) 2
   (d) 4

6. The factorization of 4x^2 - 9y^2 is:
   (a) (4x - 9y)(4x + 9y)
   (b) (2x - 3y)(2x + 3y)
   (c) (2x - 3y)(2x - 3y)
   (d) (4x - 3y)(x + 3y)

7. The expansion of (x - 2y + 3z)^2 will have how many terms?
   (a) 3
   (b) 6
   (c) 8
   (d) 9

8. If p(x) = x^2 - 2√2 x + 1, then p(2√2) is equal to:
   (a) 0
   (b) 1
   (c) 4√2
   (d) 8√2 + 1

9. The value of 102 × 98 using a suitable identity is:
   (a) 9996
   (b) 10004
   (c) 9986
   (d) 10000 - 2

10. If x + y + z = 0, then x³ + y³ + z³ is equal to:
    (a) x² + y² + z²
    (b) 3xyz
    (c) xyz
    (d) 0

Answer Key for MCQs:

1. (c) - Powers of z are 2, 1, and 0 (for the constant term √7), which are non-negative integers.
2. (b) - The term 0x^5 is zero. The highest power with a non-zero coefficient is x^4.
3. (b) - Set 3x + 5 = 0 => 3x = -5 => x = -5/3.
4. (b) - By Remainder Theorem, remainder is p(1) = (1)^3 - 1 = 1 - 1 = 0.
5. (c) - If (x + 1) is a factor, then p(-1) = 0. p(x) = 2x^2 + kx. p(-1) = 2(-1)^2 + k(-1) = 2(1) - k = 2 - k. Set 2 - k = 0 => k = 2.
6. (b) - Use a^2 - b^2 = (a - b)(a + b). Here a^2 = 4x^2 = (2x)^2 and b^2 = 9y^2 = (3y)^2. So, (2x - 3y)(2x + 3y).
7. (b) - Using (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca, which has 6 terms.
8. (b) - p(2√2) = (2√2)^2 - 2√2 (2√2) + 1 = (4 * 2) - (4 * 2) + 1 = 8 - 8 + 1 = 1.
9. (a) - 102 × 98 = (100 + 2)(100 - 2) = 100^2 - 2^2 = 10000 - 4 = 9996. (Using (a+b)(a-b)=a^2-b^2)
10. (b) - Using the identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Study these notes carefully, practice factorization, and make sure you know those identities inside out. Good luck with your preparation!