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

## 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.

## Defining Monomials

A monomial is an algebraic expression consisting of only one term. It is a product of powers of variables and a constant called the coefficient.

A single-term expression is a monomial only if all of its variables have whole numbersnon-negative and integers — as exponents. However, variables with positive exponents in the denominator are excluded because they are equivalent to a power in the numerator with the opposite exponent, according to the Quotient of Powers Property. Consider the following example.
In other words, if a variable in the denominator of an expression, it is not a monomial. The following are valid examples of monomials.
Expression Why It Is a Monomial
Any constant is a valid monomial. By the Zero Exponent Property,
The coefficient of a monomial can be
The coefficient can be negative.
A monomial can have numbers in the denominator.

Although they appear to be monomials at first glance, the single-term expressions in the following table do not satisfy the definition of a monomial.

Expression Why It Is Not a Monomial
The variables of a monomial cannot have negative integer exponents.
Monomials cannot have variables in the denominator.
The variables of a monomial must only have whole number exponents.

## Degree - Monomial

The degree of a monomial is the sum of the exponents of its variable factors. If a variable has no exponent written in it, it is assumed to be Additionally, all nonzero constants have a degree of The constant does not have a degree.

Monomial Degree
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## Identifying Monomials

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

## Calculating the Degree of a Monomial

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

## 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.
In the above example, the latter expression is not a polynomial because the term 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.
Binomial A polynomial with a exactly two terms.
Trinomial A polynomial with exactly three terms.
When there are more than three terms, the name polynomial is commonly used.

## 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
The monomial with the highest degree is This means that the degree of the polynomial is In other cases, if a polynomial consists of only a nonzero constant term, the polynomial has degree 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
Undefined
Depending on the degree of a polynomial, it can have different characteristics. For example, a polynomial with degree is linear, and a polynomial with degree is quadratic.

### Concept

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

## Identifying Polynomials

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

## 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.

## 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
It has a non-integer exponent.
It has a negative exponent.
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
Expression Number of terms

## Standard Form - Polynomial

A polynomial in one variable is expressed in standard form when the monomials that form it are arranged in decreasing degree order. This form can be represented with the following general expression.

In that general expression, is a whole number and the coefficients are real numbers. The following expression written in standard form shows a polynomial with a degree of
It should be noted that coefficients can be zero. In those cases, the corresponding terms are often omitted, which causes consecutive terms to have exponents that are not consecutive descending numbers. The following example, in standard form, shows a polynomial with a degree of
It can be seen that the term was omitted, at first. However, if chosen to do so, the polynomial can be expressed in standard form with a coefficient of and the corresponding term.

## Like Terms

Two or more terms in an algebraic expression are like terms if they have the same variable(s) with the same exponent(s).
In this expression, there are three sets of like terms — namely, terms, terms, and terms.
Note that all constants are like terms. Like terms in an expression can be combined by using addition or subtraction to rewrite the expression in simplest form.

## Combining Like Terms

In an expression, groups of like terms can be combined to write an equivalent expression with the least possible number of terms. As an example, consider the following expression.
There are three steps to follow to combine like terms.
1
Identify Like Terms
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Start by identifying the like terms in the expression. These are terms with the same variables raised to the same power. Also, all constants are like terms.
This expression has two terms, two terms, and two terms.
2
Grouping Like Terms
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Now that the like terms are identified, use the Commutative Property of Addition to rearrange the expression so that the like terms are grouped together.
3
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Finally, combine the like terms by adding or subtracting the coefficients of the variables as well as adding or subtracting the constants.
Now that all like terms have been combined, the initial expression has been reduced to its simplest form.

## Simplifying Polynomials and Identifying Their Characteristics

Izabella, so excited about solving polynomial related problems, has started a study group with Magdalena to work together on another set of polynomials. This time, the polynomials need to be written in standard form and have their characteristics identified.

Once again, they have run into a problem. The polynomials are not simplified yet. Join their study group and give them a hand.
a Simplify Polynomial I, then write it in standard form. What type of polynomial is this, according to its number of terms? Lastly, find its degree and leading coefficient.
b Simplify Polynomial II, then write it in standard form. What type of polynomial is this according to its number of terms? Lastly, find its degree and leading coefficient.

a Standard Form:

Classification: Binomial
Degree:

b Standard Form:

Classification: Trinomial
Degree:

### Hint

a A polynomial is in standard form when all of its terms are sorted by their degree in descending order.
b A polynomial of exactly one term is called a monomial. If it has two terms, then it is called a binomial and, if it has three, it is called a trinomial.

### Solution

a To simplify the polynomial, like terms will be identified and grouped.
Now, like terms will be combined by adding or subtracting their coefficients.
Note that this polynomial has exactly two terms. Therefore, it is a binomial. Next, each term will be analyzed to determine its degree and coefficient.
Term Degree Coefficient
To write the polynomial in standard form, its terms will be arranged according to their degree, in descending order.
Finally, recall that the degree and leading coefficient of a polynomial are the same as those of the monomial of highest degree. Since the highest degree monomial is the polynomial's degree is and its leading coefficient is The following table summarizes all the information found.
Standard Form
Binomial
b The same steps as in Part A will be followed. The polynomial will be simplified by identifying and combining like terms.
Note that this polynomial has exactly three terms. Therefore, it is a trinomial. Now, each term's degree and coefficient will be identified.
Term Degree Coefficient
Next, the polynomial will be written in standard form sorting its terms by degree in descending order.
Finally, since the highest degree monomial is the polynomial's degree is and its leading coefficient is The information found will be summarized using a table.
Standard Form
Trinomial

## 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.
It is always possible to rearrange these types of equations so that the right-hand side is
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 the associated equation has solutions.
Polynomial Equation Degree
Solutions
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.
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