Polynomial Functions are single-variable functions that have terms with whole number exponents. Formally, they can be written in the form:

\(f(x) = a_nx^n + a_{n-1}x^{n-1}+...a_2x^2+a_1x+ a_0\)

where \(a_0\), \(a_1\), \(a_2\) \(a_{n-1}\), \(a_n\) are real numbers called coefficients and \(n\) is a positive whole number. Typically \(a_n≠0\), because otherwise the function would just be starting from the next term. The term \(a_0\) is called the constant term.

Polynomial functions include a wide range (or families) of functions including linear, quadratic, and cubic.

Here are some rules about polynomial functions:

• Polynomial functions cannot have fraction or negative exponents
• The terms in a polynomial function cannot include multiplication or division of different variables
• Polynomial functions cannot have an exponent of \(x\)
• Polynomial functions cannot include other functions like \(\sin\) or \(log\)

You can think of a polynomial as a roller coaster shaped graph that has ups and downs. We will learn more about the shape of polynomial functions in this lesson.

---

Review

Move each function into the correct category.

---

Degree of a Polynomial

The degree of a polynomial is the highest exponent in the equation when it is written in standard form. A linear function has a degree of \(1\) because the highest exponent is \(1\) (i.e \(f(x)=3x+5\)). Notice that we don’t actually write the exponent of \(1\) in the equation because it is redundant.

Likewise, a quadratic function has a degree of \(2\) because the highest exponent is \(2\) (i.e. \(f(x) = 4x-x^2+15\)). Be careful! Sometimes the term with the highest exponent is not written first! Find the term with the highest exponent to identify the degree of the polynomial.

When an equation is written in factored form, we need to expand and convert it to standard form to identify the degree of the polynomial function. For example, the equation \(f(x)=(x-2)(x+1)\) appears to have a degree of \(1\) because the highest exponent is \(1\). However, it can be expanded to \(f(x) = x^2-x-2\) and we can see that the highest exponent is \(2\). Therefore, the degree of the polynomial function \(f(x)=(x-2)(x+1)\) is \(2\).

We can look at the base (or parent) functions for the first 6 degree polynomials:

It should be noted that the red graphs represent odd polynomials while the blue graphs represent even polynomials.

---

Leading Coefficient of a Polynomial

The ‘\(a\)’ terms of a polynomial equation are called coefficients. The leading coefficient is the coefficient of the term with the highest degree (exponent of \(n\)). The sign (positive or negative) of the leading coefficient tells us information about the polynomial. A negative leading coefficient results in a reflection about the \(x\)-axis (a vertical reflection). The leading coefficient impacts the functions “end behaviour”.

The leading coefficients for the linear functions below are \(+3\) (left) and \(-3\) (right). When the leading coefficient becomes negative, the graph is reflected about the \(x\)-axis.

The leading coefficients for the quadratic functions below are \(+1\) (left) and \(-1\) (right). Again, the graph is reflected about the \(x\)-axis. But, for the quadratic functions, this reflection also changes extreme values. When the leading coefficient is positive, the function has a minimum. When the leading coefficient is negative, the function has a maximum.

---