body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} You must have JavaScript enabled to use this site.

Recommended

Tests

An error ocurred, try again later!

eCourses /

Algebra 1 /

Chapter {{ article.chapter.number }}

{{ article.number }}.

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount}}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount}}
	{{ 'ml-lesson-time-estimation' \| message }}

Image Credits

{{ item.file.title }}
{{ presentation }}

No file copyrights entries found

Various events in daily life depend on multiple factors. Consider the

CO_{2}

emission of a car. It depends on many variables, such as the gas used, the engine size, and the number of cylinders. Countless daily events like that can be modeled using a combination of powers of different variables. Making those models requires using algebraic expressions with multiple terms called polynomials.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Variable
Algebraic Expressions
Term
Powers
Exponents
Zero Exponent Property

Explore

Classifying Algebraic Expressions

Take a look at the following single-term algebraic expressions. Can a shared characteristic be recognized which would allow for them to be classified reasonably? Drag the expressions and use the colored areas to visualize any common themes, and identify which expressions could be grouped together.

Interactive applet allowing to drag expressions to classify them

Discussion

Defining Monomials

Pop Quiz

Identifying Monomials

The next applet shows different algebraic expressions. Use the new knowledge introduced previously to determine whether or not these expressions are monomials.

Interactive graph showing different algebraic expressions

Pop Quiz

Calculating the Degree of a Monomial

The following applet shows different monomials. Determine the given monomial's degree.

Interactive applet showing different monomials

Discussion

Defining Polynomials

A polynomial is either a single monomial or a sum of them. Each of the monomials that form the polynomial are referred to as a term of the polynomial. This definition implies that, for an expression to be a polynomial, each of its terms must be a valid monomial.

Example Polynomial ✓ 5 x y + 2 x y^{2} + 8 x - 9 Not a Polynomial \times 3 x y + 2 x y^{- 2} + 7 x y^{3} - 1

In the above example, the latter expression is not a polynomial because the term

2 x y^{- 2}

is not a monomial. The variables in a monomial can only have whole numbers as exponents.

Classifying Polynomials by Their Number of Terms

One way that polynomials can be classified is according to the number of terms they have. The following table shows the names used for this classification.

Name	Definition	Example
Monomial	A polynomial with a single term.	$3 x^{2} y^{3}$
Binomial	A polynomial with a exactly two terms.	$5 x y + 3 x^{2} y^{3}$
Trinomial	A polynomial with exactly three terms.	$x^{3} - 9 x + 4$

When there are more than three terms, the name polynomial is commonly used.

Concept

Degree - Polynomial

The degree of a polynomial is defined as the greatest of the degrees of its terms. For example, the following is a polynomial of degree

5 .

Interactive graphs showing the degreee of each monomial of the polynomial 3xy+4xy^4-9xy^2-84

The monomial with the highest degree is

4 x y^{4} .

This means that the degree of the polynomial is

1 + 4 = 5 .

In other cases, if a polynomial consists of only a nonzero constant term, the polynomial has degree

0 .

For the special case of the zero polynomial, the degree is undefined. The following table summarizes this information by providing some specific examples.

Polynomial	Degree
$- 5 x^{2} y + x^{2} y^{4} - 11 x^{2} - 3$	$6$
$x - 11 x^{4} + 8 x^{3}$	$4$
$7$	$0$
$0$	Undefined

Depending on the degree of a polynomial, it can have different characteristics. For example, a polynomial with degree

1

is linear, and a polynomial with degree

2

is quadratic.

Concept

Leading Coefficient

In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Pop Quiz

Identifying Polynomials

The following applet alternates between different algebraic expressions. Determine whether or not these expressions are polynomials.

Interactive applet showing different algebraic expressions

Pop Quiz

Identifying a Polynomial's Degree and Leading Coefficient

The following applet alternates between different polynomials. Determine the given polynomial's degree or leading coefficient as indicated.

Interactive applet showing different polynomials

Example

Identifying Polynomials and Their Number of Terms

Izabella is improving her understanding of polynomials in her algebra course. However, she has come across some problems where she needs to identify the number of terms of some algebraic expressions and only some of the expressions are polynomials.

Izabella feels confused. Help her solve these problems.

Hint

For an algebraic expression to be a polynomial each of its terms must be a monomial. Therefore, exponents of the variables involved can only be whole numbers.

Solution

For an algebraic expression to be a polynomial, each term must be a valid monomial. A monomial is a single-term expression that is a product of numbers and variables raised to exponents that are whole numbers. Three of the given expressions do not satisfy this definition. That takes them out of the running as being polynomials.

Expression	Reason
$4 x + x^{\frac{1}{2}}$	It has a non-integer exponent.
$2 x^{2} + 5 + y^{- 2}$	It has a negative exponent.
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	It has a variable in the denominator.

Now that Izabella knows which expressions are polynomials and which are not, she can figure out how to identify their number of terms. Recall that, in any algebraic expression, terms are separated by a plus or a minus sign.

Polynomial	Number of terms
$\frac{x}{4} - 3 x^{3}$	$2$
$2 x^{2} + 5 - 7 y$	$3$
$\frac{x y}{8} + 3 x^{3} - 3 y^{3} + 2 x$	$4$
Expression	Number of terms
$4 x + x^{\frac{1}{2}}$	$2$
$2 x^{2} + 5 + y^{- 2}$	$3$
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	$4$

Closure

Introduction to Polynomial Equations

Now that polynomials have been introduced, equations containing them can be managed. The following example shows a polynomial equation, also known as an algebraic equation.

2 x^{3} - 12 = - 8 x^{2} - 2 x

It is always possible to rearrange these types of equations so that the right-hand side is

0 .

2 x^{3} + 8 x^{2} + 2 x - 12 = 0

In this format, the degree of the polynomial in the left-hand side defines the degree of the equation. It is a remarkable fact that the number of solutions of a polynomial equation is equal to its degree. This relationships is known as the Fundamental Theorem of Algebra. Since the example polynomial equation above is of degree

3,

the associated equation has

3

solutions.

Polynomial Equation	Degree
$2 x^{3} + 8 x^{2} + 2 x - 12 = 0$	$3$
Solutions
$x = - 3, x = - 2, x = 1$

Polynomial equations can be solved algebraically by using methods such as the Quadratic Formula or others based on the Zero Product Property. Alternatively, they can also be solved graphically or by using numerical methods.

{{ article.displayTitle }}

Catch-Up and Review

Classifying Polynomials by Their Number of Terms

Hint

Solution