MA001: College Algebra

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 of power, or in general form. 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 highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

TERMINOLOGY OF POLYNOMIAL FUNCTIONS

We often rearrange polynomials so that the powers are descending.

When a polynomial is written in this way, we say that it is in general form.

HOW TO

Given a polynomial function, identify the degree and leading coefficient.

EXAMPLE 5

Identifying the Degree and Leading Coefficient of a Polynomial Function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

$\begin{aligned} f(x) &=3+2 x^{2}-4 x^{3} \\ g(t) &=5 t^{5}-2 t^{3}+7 t \\ h(p) &=6 p-p^{3}-2 \end{aligned}$

Solution

For the function $f(x)$, the highest power of $x$ is $3$, so the degree is $3$. The leading term is the term containing that degree, $-4 x^{3}$. The leading coefficient is the coefficient of that term, $−4$.

For the function $g(t)$, the highest power of $t$ is $5$, so the degree is $5$. The leading term is the term containing that degree, $5 t^{5}$. The leading coefficient is the coefficient of that term, $5$.

For the function $h(p)$, the highest power of $p$ is $3$, so the degree is $3$. The leading term is the term containing that degree, $-p^{3}$. The leading coefficient is the coefficient of that term, $−1$.

TRY IT #3

Identify the degree, leading term, and leading coefficient of the polynomial $f(x)=4 x^{2}-x^{6}+2 x-6$.