Mathematics: Class 10 — Chapter 2: Polynomials

• Polynomial: An algebraic expression in one or more variables with non-negative integer exponents.
• Degree of a Polynomial: The highest power of the variable x in the polynomial p(x).
• Linear Polynomial: A polynomial of degree 1. General form: ax + b (where a ≠ 0).
• Quadratic Polynomial: A polynomial of degree 2. General form: ax² + bx + c (where a ≠ 0). Derived from "quadrate" meaning square.
• Cubic Polynomial: A polynomial of degree 3. General form: ax³ + bx² + cx + d (where a ≠ 0).

• Value of p(x) at x = k: Obtained by replacing x with a real number k, denoted as p(k).
• Zero of a Polynomial: A real number k is a zero of a polynomial p(x) if p(k) = 0.
• Linear Zero Formula: For p(x) = ax + b, the zero is k = -b/a, which is equal to:
  - (Constant term) / (Coefficient of x)

• The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
• Linear Polynomial Graph: A straight line that intersects the x-axis at exactly one point: (-b/a, 0).
• Quadratic Polynomial Graph: A curve called a parabola. It opens upwards (∪) if a > 0 and downwards (∩) if a < 0.
• Quadratic Zeroes Cases:
  • Case (i): Intersects x-axis at two distinct points (2 zeroes).
  • Case (ii): Touches x-axis at exactly one point (1 zero / coincident points).
  • Case (iii): Completely above or below the x-axis (no real zeroes).
• General Rule: A polynomial of degree n can intersect the x-axis at at most n points. Therefore, a polynomial of degree n has at most n zeroes.

If α and β are the zeroes of the quadratic polynomial ax² + bx + c (a ≠ 0):

• Sum of Zeroes (α + β):
  α + β = -b/a = - (Coefficient of x) / (Coefficient of x²)
• Product of Zeroes (αβ):
  αβ = c/a = (Constant term) / (Coefficient of x²)

If α, β, and γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d (a ≠ 0):

• Sum of Zeroes:
  α + β + γ = -b/a
• Sum of Products taken two at a time:
  αβ + βγ + γα = c/a
• Product of Zeroes:
  αβγ = -d/a