Polynomials

Types of Polynomials

• Monomials: Polynomials with only one term, e.g. 3x^2, 5y
• Binomials: Polynomials with two terms, e.g. x^2 + 3x, 2y - 5
• Trinomials: Polynomials with three terms, e.g. x^2 + 2x + 1, 3y^2 - 2y + 1
• Monic Polynomials: Polynomials with leading coefficient 1, e.g. x^2 + 2x + 1, y^3 - 2y^2 + 1

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable, e.g.:
  • Degree of x^2 + 3x - 1 is 2
  • Degree of y^3 - 2y^2 + 1 is 3
  • Degree of 2x - 3 is 1

Zeroes of a Polynomial

• A zero or root of a polynomial is a value of the variable that makes the polynomial equal to zero, e.g.:
  • Zeroes of x^2 - 4 are x = 2 and x = -2
  • Zeroes of y^3 - 2y^2 - 5y + 6 are y = 1, y = 2, and y = 3

Remainder Theorem

• If a polynomial f(x) is divided by (x - a), the remainder is f(a), e.g.:
  • If f(x) = x^2 + 2x - 3 and a = 2, then the remainder is f(2) = 2^2 + 2(2) - 3 = 5

Factor Theorem

• If x - a is a factor of a polynomial f(x), then a is a zero of f(x), and vice versa, e.g.:
  • If x - 2 is a factor of f(x), then 2 is a zero of f(x)

Factorisation

• Factorisation is the process of expressing a polynomial as a product of simpler polynomials, e.g.:
  • x^2 - 4 = (x - 2)(x + 2)
  • y^3 - 2y^2 - 5y + 6 = (y - 1)(y - 2)(y - 3)
• Factor trees can be used to help with factorisation, e.g.:
  • x^2 + 5x + 6 = (x + 3)(x + 2)