Higher-degree polynomial functions (Higher Tier)

Understanding Higher-degree Polynomial Functions

A higher-degree polynomial function is a polynomial with a degree greater than 2. The degree of a polynomial refers to the highest exponent in the polynomial expression.
The most common forms of higher-degree polynomial functions you will encounter are cubic (degree 3) and quartic (degree 4) functions.
Like all polynomials, higher-degree polynomial functions can have real-valued coefficients and variables.
A polynomial of degree n typically has n roots or solutions, which can be real or complex numbers. For example, a cubic function has three roots, and a quartic function has four roots.

Graphing Higher-degree Polynomial Functions

The graph of a polynomial function of degree n is a smooth curve that has at most n-1 ‘bends’ or turning points, also known as extrema.
The end behaviour of a polynomial function’s graph is determined by its degree and leading coefficient. If the degree is odd and the leading coefficient is positive, the graph extends from the third quadrant to the first quadrant. If the degree is odd and the leading coefficient is negative, the graph extends from the second quadrant to the fourth quadrant. If the degree is even, the graph extends from the second quadrant to the first quadrant if the leading coefficient is positive, and from the third quadrant to the fourth quadrant if the leading coefficient is negative.
You can use synthetic division or the Rational Root Theorem to find the roots of a polynomial, which are the x-values where the graph crosses the x-axis.

Factoring Higher-degree Polynomial Functions

Factoring is the process of breaking down a polynomial into simpler terms, or factors, that multiply together to give the original polynomial.
The Factor Theorem is a crucial tool for factoring higher-degree polynomials. It states that a polynomial f(x) has a factor x-k if and only if f(k) = 0. In other words, k is a root of the polynomial.
In order to factor a higher-degree polynomial completely, you must find all its roots, then express the polynomial as a product of linear factors (and possibly a quadratic factor if the polynomial has a degree of 3 or higher and complex roots).

Solving Higher-degree Polynomial Equations

To solve higher-degree polynomial equations, you can use factoring, the Quadratic Formula (if the polynomial is of degree 2), the Rational Root Theorem, and synthetic division.
If a polynomial cannot be factored, you can use numerical methods such as the Newton-Raphson method to approximate its roots.
Remember, a polynomial equation of degree n has exactly n complex roots, counting multiplicity. These roots can be real or non-real. The Fundamental Theorem of Algebra guarantees this. If a complex number is a root, so is its complex conjugate.