CBSE Class 10th Maths : Chapter 2 – Polynomials and Formulas 

Formulas in Chapter 2

Polynomials, the building blocks of algebra, are essential for your CBSE Class 10 Maths journey. This guide delves into all the important formulas of Chapter 2, providing detailed explanations and examples to solidify your grasp on these concepts.

Expressions consisting of variables (represented by letters like x, y, z) and coefficients (numerical constants), combined using addition, subtraction, and multiplication. (Examples: x² + 3x – 4, 2y, 5)

Formulas for Basic Operations

Addition, subtraction, multiplication of polynomials following the order of operations (PEMDAS).

1. Addition: Arrange the polynomials by descending degree (highest power of the variable first) and combine like terms (terms with the same variable raised to the same power).
   
   • Example 1: Add (3x² + 2x – 1) and (2x² – x + 5).
     
     • Explanation: Arrange terms: 3x² + 2x – 1
       • (2x² – x + 5)
     • Combine like terms: (3 + 2)x² + (2 – 1)x + (-1 + 5) = 5x² + x + 4.
   • Example 2: Add (x³ – 2x² + 5x) and (-3x³ + x² – 7).
     
     • Explanation: Even though terms with different exponents exist, they’re not “like” terms. Arrange and add directly: x³ + (-2 + 1)x² + (5 – 7)x = x³ – x² – 2x.
2. Subtraction: Arrange the polynomials as with addition and subtract corresponding terms.
   
   • Example 1: Subtract (x² – 3x + 2) from (2x² + 5x – 1).
     
     • Explanation: Arrange terms: 2x² + 5x – 1
       • (x² – 3x + 2)
     • Subtract terms: (2 – 1)x² + (5 + 3)x + (-1 – 2) = x² + 8x – 3.
   • Example 2: Subtract (4y + 3) from (2y² – y + 5).
     
     • Explanation: Notice the absence of a y² term in the first polynomial. Add a 0y² term for proper alignment: 2y² – y + 5
       • (0y² + 4y + 3)
     • Subtract terms: 2y² + (-1 – 4)y + (5 – 3) = 2y² – 5y + 2.

Formulas for Factorization

Breaking down polynomials into simpler expressions. We’ll focus on factoring by grouping and using identities like a² – b² = (a + b)(a – b).

• Example: Factorize x² + 5x + 6.
  • Explanation: This can be factored by grouping. We need to find two numbers that add up to 5 and multiply to 6. (2 and 3 fit the criteria). The expression becomes (x + 2)(x + 3).

1. Factoring by Grouping: This method is used for polynomials in the form ax² + bx + c. Find two values (p and q) that satisfy:
   
   • p + q = b (coefficient of x term)
     
   • pq = ac (product of the first and last term coefficients)
     
   • Example 1: Factor 3x² + 11x + 4.
     • Explanation: Find p and q such that p + q = 11 and pq = (3)(4) = 12. We see p = 4 and q = 3 satisfy both conditions. Rewrite the bx term using p and q, group, and factor: 3x² + (4 + 3)x + 4x + (4)(3) = 3x² + 7x + 4x + 12 = (3x + 4)(x + 3)
   • Example 2: Factor 2y² – 7y + 6.
     • Explanation: Find p and q such that p + q = -7 and pq = (2)(6) = 12. We see p = -3 and q = -4 satisfy both conditions. Follow the same process as the previous example to get: (2y – 3)(y – 2).
2. Factoring using Identities: These pre-established formulas help factor specific polynomial forms directly.
   
   • Difference of Squares: a² – b² = (a + b)(a – b)
     
     • Example: Factor x² – 9.
       • Explanation: Recognize the pattern a² (x²) and b² (9, which is 3²). Apply the formula: x² – 9 = (x + 3)(x – 3).
   • Perfect Square Trinomial: (a + b)² = a² + 2ab + b²
     
     • Example: Factor (x + 2)².
       • Explanation: Identify a (x) and b (2) in the given expression
     • Factoring using Identities (continued):
       
   • Sum or Difference of Cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)
     • Example: Factor 8x³ + 1.
       • Explanation: Recognize the pattern a³ (8x³) and b³ (1, which is 1³). As it’s a sum of cubes, use the appropriate formula with a = 2x and b = 1: 8x³ + 1 = (2x + 1)(4x² – 2x + 1).

Formulas for Zeroes of a Polynomial

1. Zeroes of a Polynomial: The values of the variable (x) that make the polynomial equal to zero. You can find them by setting the polynomial equal to zero and solving the resulting equation.
   
   • Example: Find the zeroes of the polynomial 2x² + 5x – 3.
     • Explanation: Set the polynomial equal to zero and solve the resulting equation: 2x² + 5x – 3 = 0. This might involve factoring or using other algebraic methods to find x = -3/2 and x = 1.

Short Notes

• Understanding the logic behind the formulas is crucial. Memorization alone might not be sufficient in the long run.
• Practice applying these formulas to various polynomials to solidify your understanding and develop problem-solving skills.
• Don’t hesitate to refer to your textbook or ask your teacher for clarification on any concepts.
• These formulas are tools for manipulating polynomials. The ultimate goal is to understand the underlying algebraic relationships they represent.

Practice Questions (Chapter 2)

1. Add the polynomials: 2x² + 3x – 1 and x² – 2x + 5.
2. Factorize the polynomial: x² – 7x + 12.
3. Find the zeroes of the polynomial: 3x² – 5x + 2.

By mastering these formulas and practicing regularly, you’ll be well-equipped to tackle polynomial problems with confidence in your CBSE Class 10 Maths exam