Defining Polynomials

  • A polynomial function is a function comprising of several terms. These terms are either constants, or a constant multiple of an integer power of a variable.
  • Each term in a polynomial function is called a monomial, and the constant or integer is known as the coefficient.
  • In the polynomial equation, the highest power of the variable identifies the degree of the polynomial.
  • The term with the highest degree is known as the leading term, while the coefficient of the leading term is the leading coefficient.

Understanding Polynomial Graphs

  • The graphs of polynomial functions are continuous, and without any sharp bends or gaps.
  • Understanding how to plot polynomial graphs involves knowing the degree and the leading coefficient. This helps identify the basic shape of the graph.
  • A polynomial of odd degree will have ends that point in opposite directions.
  • A polynomial of even degree will have ends that point in the same direction.

Division of Polynomials

  • The Factor theorem provides a quick means of checking whether (x-a) is a factor of a polynomial, P(x). If P(a) = 0, then x = a is a root of the polynomial and (x-a) is a factor.
  • Long division of polynomials can be utilised when dividing a polynomial by a factor of a higher degree.
  • If a polynomial f(x) is divided by (x - a) and the remainder is zero, then ‘a’ is said to be a root of the equation f(x) = 0.

Polynomial Inequalities

  • To solve polynomial inequalities, first solve the equation obtained by setting the polynomial greater than or equal to zero, and then determine which of the intervals defined by these solutions satisfy the original inequality.

Rational Root Theorem

  • The Rational Root Theorem is an important theorem in polynomial equations. It states that if a polynomial has rational roots or zeros, then they are a fraction derived from the ratio of the factors of the constant term to the factors of the leading coefficient.

Fundamental Theorem of Algebra

  • The Fundamental Theorem of Algebra asserts that each polynomial equation of degree n has precisely n complex roots or zeros, including repeated roots.