# Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a**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 [latex]384\pi [/latex], is known as a

**coefficient**. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]{a}_{i}{x}^{i}[/latex], such as [latex]384\pi w[/latex], 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 [latex]5{x}^{4}[/latex], is called a

**monomial**. A polynomial containing two terms, such as [latex]2x - 9[/latex], is called a

**binomial**. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[/latex], 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

[latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

Each real number *a*is called a

_{i}**coefficient**. The number [latex]{a}_{0}[/latex] that is not multiplied by a variable is called a

*constant*. Each product [latex]{a}_{i}{x}^{i}[/latex] 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.- [latex]3+2{x}^{2}-4{x}^{3}\\[/latex]
- [latex]5{t}^{5}-2{t}^{3}+7t[/latex]
- [latex]6p-{p}^{3}-2[/latex]

### Solution

- The highest power of
*x*is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[/latex]. The leading coefficient is the coefficient of that term, [latex]-4[/latex]. - The highest power of
*t*is [latex]5[/latex], so the degree is [latex]5[/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[/latex]. The leading coefficient is the coefficient of that term, [latex]5[/latex]. - The highest power of
*p*is [latex]3[/latex], so the degree is [latex]3[/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[/latex], The leading coefficient is the coefficient of that term, [latex]-1[/latex].

### Try It 1

Identify the degree, leading term, and leading coefficient of the polynomial [latex]4{x}^{2}-{x}^{6}+2x - 6[/latex]. Solution## Licenses & Attributions

### CC licensed content, Specific attribution

- College Algebra.
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