Concept of Polynomials

Introduction

• Chapter Number 2: Polynomials
• Concept of polynomials and their utility
• Preparation for NCERT questions

Basic Terms

Variables and Constants

• Variable: Values that change. Example: x, y, z, etc.
• Constant: Values that are fixed. Example: 2, -1, π, etc.

Algebraic Expression

• Combination of terms using plus and minus
• For example, 2x - 3y + 4

Polynomials

• Algebraic expression where the variable's power is a whole number. Example: 3y² - 2y + 1

Degree and Terms

• Degree of Polynomials: The highest power of the variable available
• Terms: Combination of variables and constants

Types of Polynomials

Based on Number of Terms

• Monomial: One term (e.g. 6x)
• Binomial: Two terms (e.g. 6x + 2)
• Trinomial: Three terms (e.g. 6x² - 4x + 2)

Based on Degree

• Linear: Degree = 1 (e.g. 2x + 1)
• Quadratic: Degree = 2 (e.g. 3x² - 4x + 2)
• Cubic: Degree = 3 (e.g. x³ - 3x² + 3x - 1)

Remainder Theorem and Factor Theorem

Remainder Theorem

• When a polynomial Px is divided by X - a, the remainder is equal to P(a).

Factor Theorem

• If Px divided by X - a gives a remainder of 0, then X - a is a factor of Px.
• X - a is a factor if P(a) = 0. The converse is also true.

Zeros and Roots of Polynomials

• Zeros of Polynomials: Values of P(x) that make P(x) = 0.

Value of Polynomials

• Value of Polynomials: Substitute the given value in place of x in Px and solve.

Algebraic Identities

• Some important identities:

  • (a + b)² = a² + b² + 2ab
  • (a - b)² = a² + b² - 2ab
  • a² - b² = (a + b)(a - b)
  • (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
  • (a + b)³ = a³ + b³ + 3ab(a + b)
  • (a - b)³ = a³ - b³ - 3ab(a - b)

• Using compression and expansion formulas

Practice

• Do NCERT exercise 2.3 at home
• Solve additional questions from RD Sharma and RS Aggarwal books

Conclusion

• The concept of polynomials is extensive and important
• Proper use of Remainder and Factor Theorem
• Practice will solve all questions