Polynomial Functions: Learn It 1

  • Identify the degree and leading coefficient of polynomial functions.
  • Identify end behavior of polynomial functions.
  • Identify intercepts of factored polynomial functions.

Polynomial Functions

Polynomial functions allow us to describe curves with multiple peaks and valleys, making them perfect for capturing the intricacies of real-world data.

polynomial functions

Let [latex]n[/latex] be a non-negative integer. A polynomial function is a function that can be written in the form

 

[latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

 

This is called the general form of a polynomial function. Each [latex]{a}_{i}[/latex] is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[/latex] is a term of a polynomial function.

Which of the following are polynomial functions?

[latex]\begin{array}{c}f\left(x\right)=2{x}^{3}\cdot 3x+4\hfill \\ g\left(x\right)=-x\left({x}^{2}-4\right)\hfill \\ h\left(x\right)=5\sqrt{x}+2\hfill \end{array}[/latex]

Degree and Leading Coefficient of a Polynomial Function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order based on the power on the variable. This is called writing a polynomial in general or standard form.

terminology of a polynomial function

  • The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.
  • The leading term is the term containing the variable with the highest power, also called the term with the highest degree.
  • The leading coefficient is the coefficient of the leading term.

Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +…+a_2x^2+a_1x+a_0.

How To: Given a polynomial function, identify the degree and leading coefficient

  1. Find the highest power of [latex]x[/latex]to determine the degree of the function.
  2. Identify the term containing the highest power of [latex]x[/latex]to find the leading term.
  3. The leading coefficient is the coefficient of the leading term.
Identify the degree, leading term, and leading coefficient of the following polynomial functions.

[latex]\begin{array}{l} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\h\left(p\right)=6p-{p}^{3}-2\end{array}[/latex]