# Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a

A polynomial containing only one term, such as

We can find the

Each real number

Solution

**polynomial**, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as$384\pi$

, is known as a **coefficient**. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product${a}_{i}{x}^{i}$

, such as $384\pi w$

, is a **term of a polynomial**. If a term does not contain a variable, it is called a*constant*.A polynomial containing only one term, such as

$5{x}^{4}$

, is called a **monomial**. A polynomial containing two terms, such as$2x - 9$

, is called a **binomial**. A polynomial containing three terms, such as$-3{x}^{2}+8x - 7$

, is called a **trinomial**.We can find the

**degree**of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the**leading term**because it is usually written first. The coefficient of the leading term is called the**leading coefficient**. When a polynomial is written so that the powers are descending, we say that it is in standard form.### A General Note: Polynomials

A**polynomial**is an expression that can be written in the form

${a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

*a*is called a

_{i}**coefficient**. The number

${a}_{0}$

that is not multiplied by a variable is called a *constant*. Each product

${a}_{i}{x}^{i}$

is a **term of a polynomial**. The highest power of the variable that occurs in the polynomial is called the

**degree**of a polynomial. The

**leading term**is the term with the highest power, and its coefficient is called the

**leading coefficient**.

### How To: Given a polynomial expression, identify the degree and leading coefficient.

- Find the highest power of
*x*to determine the degree. - Identify the term containing the highest power of
*x*to find the leading term. - Identify the coefficient of the leading term.

### Example 1: Identifying the Degree and Leading Coefficient of a Polynomial

For the following polynomials, identify the degree, the leading term, and the leading coefficient.- $3+2{x}^{2}-4{x}^{3}\\$
- $5{t}^{5}-2{t}^{3}+7t$
- $6p-{p}^{3}-2$

### Solution

- 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,$-4$. - 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,$5$. - 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 1

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

.Solution