2. Polynomials

  • 2

    Polynomials

    This section introduces polynomials, exploring their definitions, types, degree, and the concepts of zeroes and factorization.

  • 2.1

    Introduction

    This section introduces polynomials, their key characteristics, and significant theorems related to polynomial factorization.

  • 2.2

    Polynomials In One Variable

    The section provides an overview of polynomials in one variable, introducing key definitions, types, and properties, including degrees and coefficients.

  • 2.3

    Zeroes Of A Polynomial

    This section introduces the concept of zeroes of a polynomial, explaining how to find them and their significance.

  • 2.4

    Factorisation Of Polynomials

    This section introduces the concept of factorisation of polynomials and the Factor Theorem, illustrating how to identify polynomial factors based on their roots.

  • 2.5

    Algebraic Identities

    Algebraic identities are fundamental equations that hold true for any value of their variables.

  • 2.6

    Summary

    This section outlines key concepts related to polynomials, including definitions, terms, and important theorems.

    • Key Summary

    Polynomials represent a significant class of algebraic expressions formed by combining variables, constants, and non-negative integer exponents. The chapter elaborates on various types of polynomials, their classifications based on degrees, the concepts of zeros and factors, and the application of algebraic identities in factorization. It emphasizes the importance of the Remainder and Factor Theorems in understanding polynomials in one variable as well as in multiple variables.

    Key Takeaways

    • A polynomial in one variable is an algebraic expression of the form p(x) = anxn + an–1xn–1 + ... + a2x2 + a1x + a0.
    • Polynomials are classified into monomials, binomials, and trinomials based on the number of terms.
    • Each polynomial has a degree which indicates the highest power of the variable in the polynomial.

    Key Concepts

    • Polynomial: An algebraic expression consisting of variables raised to non-negative integer powers, combined with coefficients.
    • Degree of a Polynomial: The highest exponent of the variable in the polynomial.
    • Zero of a Polynomial: A value for which the polynomial evaluates to zero.
    • Factor Theorem: States that if p(a) = 0, then (x - a) is a factor of the polynomial p(x).
    • Algebraic Identities: Equations that hold true for all values of the variables in them, such as (x + y)² = x² + 2xy + y².