Polynomials (Class 9 Mathematics)

• Introduction to Polynomials

  • Definition: Algebraic expressions with non-negative integer exponents of variables
  • Variables: Symbols like $x$, $y$, $z$ representing unknown quantities
  • Coefficients: Numerical factors attached to variables
  • Algebraic Identities vs. Polynomials

• Degree of a Polynomial

  • Definition: Highest power of the variable in a polynomial
  • Linear Polynomial: Degree $1$ (e.g., $ax + b$)
  • Quadratic Polynomial: Degree $2$ (e.g., $ax^2 + bx + c$)
  • Cubic Polynomial: Degree $3$ (e.g., $ax^3 + bx^2 + cx + d$)
  • Constant Polynomial: Degree $0$ (e.g., $5$)
  • Zero Polynomial: Zero itself (Degree is not defined)

• Types of Polynomials

  • Based on Terms:
    • Monomial: Exactly $1$ term (e.g., $5x$)
    • Binomial: Exactly $2$ terms (e.g., $x + 5$)
    • Trinomial: Exactly $3$ terms (e.g., $x^2 + 2x + 1$)
  • Based on Degree:
    • Linear, Quadratic, Cubic

• Zeros of a Polynomial

  • Definition: Value of the variable for which the polynomial becomes zero
  • Finding the zero of $p(x)$: Solve $p(x) = 0$
  • Geometric meaning: X-coordinate of the point where the graph intersects the x-axis
  • Remainder Theorem:
    • Statement: If $p(x)$ is divided by $(x - a)$, the remainder is $p(a)$
  • Factor Theorem:
    • Statement: If $p(a) = 0$, then $(x - a)$ is a factor of $p(x)$, and vice versa

• Factorisation of Polynomials

  • Splitting the Middle Term: Used for quadratic polynomials ($ax^2 + bx + c$)
  • Factorisation using Factor Theorem: Used for cubic and higher-degree polynomials

• Algebraic Identities

  • $(x + y)^2 = x^2 + 2xy + y^2$
  • $(x - y)^2 = x^2 - 2xy + y^2$
  • $x^2 - y^2 = (x + y)(x - y)$
  • $(x + a)(x + b) = x^2 + (a + b)x + ab$
  • $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$
  • $(x + y)^3 = x^3 + y^3 + 3xy(x + y)$
  • $(x - y)^3 = x^3 - y^3 - 3xy(x - y)$
  • $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$