# Polynomials

#### 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.