Polynomials

• Polynomial


  A mathematical expression involving one or more variables and coefficients, and operations such as addition, subtraction, multiplication, and non-negative integer exponents
• Polynomial


  • Highest power of variable is called the degree of the polynomial
  • Examples: 4x + 2 (degree 1), 2y^2 - 3y + 4 (degree 2), 5x^3 - 4x^2 + x - √√2 (degree 3), 7u^6 - 3u^3 + 4u^2 + u (degree 6)
• Linear polynomial


  A polynomial of degree 1
• Quadratic polynomial


  A polynomial of degree 2
• Cubic polynomial


  A polynomial of degree 3
• Cubic polynomials


  • 2 - 2√3x^3 + 3x^2 - 2x + 1
• Value of a polynomial p(x) at x=k


  The value obtained by replacing x with k in p(x)
• p(-1) = 0 and p(4) = 0 means -1 and 4 are zeroes of the quadratic polynomial p(x) = x^2 - 3x - 4
• Zero of a polynomial p(x)

  A real number k such that p(k) = 0
• Finding zeroes of a linear polynomial p(x) = ax + b

  1. p(k) = ak + b = 0

  2. k = -b/a
• The zero of a linear polynomial ax + b is related to its coefficients a and b
• The zeroes of a quadratic polynomial are also related to its coefficients
• For a quadratic polynomial p(x) = ax^2 + bx + c

  • Sum of zeroes = -b/a
  • Product of zeroes = c/a
• For a cubic polynomial p(x) = ax^3 + bx^2 + cx + d

  • Sum of zeroes = -b/a
  • Sum of products of zeroes taken two at a time = -c/a
  • Product of zeroes = -d/a
• Polynomial


  Algebraic expression of the form ax^n + bx^(n-1) + ... + c, where a, b, c are real numbers and n is a non-negative integer
• Types of polynomials


  • Linear (degree 1)
  • Quadratic (degree 2)
  • Cubic (degree 3)
• Quadratic polynomial


  • ax^2 + bx + c, where a, b, c are real numbers and a ≠ 0
• Zeros of a polynomial

  • The x-coordinates of the points where the graph of the polynomial intersects the x-axis
• A quadratic polynomial can have at most two zeros
• A cubic polynomial can have at most three zeros
• Sum of the zeros of a quadratic polynomial

  • b/a, where a, b are the coefficients of the quadratic polynomial ax^2 + bx + c
• Product of the zeros of a quadratic polynomial

  c/a, where a, c are the coefficients of the quadratic polynomial ax^2 + bx + c
• Finding the factors of a quadratic polynomial

  If α and β are the zeros of the quadratic polynomial p(x) = ax^2 + bx + c, a ≠ 0, then x - α and x - β are the factors of p(x)
• Sum of the zeros of a cubic polynomial

  • b/a, where a, b, c, d are the coefficients of the cubic polynomial ax^3 + bx^2 + cx + d
• If x + y + z = 0, then x^2 + y^2 + z^2 = 3xyz
• Geometrical meaning of the zeros of a polynomial

  The x-coordinates of the points where the graph of the polynomial intersects the x-axis
• Graphs of polynomials


  • The number of zeros is equal to the number of points where the graph intersects the x-axis