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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.
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.
Discussion
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 numbers — non-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. 5x/y^2 = 5xy^(- 2) 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 |
|---|---|
| 5 | Any constant is a valid monomial. By the Zero Exponent Property, 5x^0=5. |
| 0 | The coefficient of a monomial can be 0. |
| - 2x^5 | The coefficient can be negative. |
| x^3y/5 | 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 |
|---|---|
| 2x^(- 1) | The variables of a monomial cannot have negative integer exponents. |
| 4x^3/y | Monomials cannot have variables in the denominator. |
| 5 x^3y^(12) | The variables of a monomial must only have whole number exponents. |
Concept
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 1. Additionally, all nonzero constants have a degree of 0. The constant 0 does not have a degree.
| Monomial | Degree |
|---|---|
| 3x | 1 |
| x^2 | 2 |
| 9x^3 | 3 |
| x^3y | 4 |
| 7 | 0 |
| a^3b^4c^5/13 | 12 |
| 0 | undefined |
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.
Pop Quiz
Calculating the Degree of a Monomial
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 ✓ 5xy+2xy^2+8x-9 Not a Polynomial * 3xy+2xy^(- 2)+7xy^3-1 In the above example, the latter expression is not a polynomial because the term 2xy^(-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. | 3x^2y^3 |
| Binomial | A polynomial with a exactly two terms. | 5xy + 3x^2y^3 |
| Trinomial | A polynomial with exactly three terms. | x^3-9x+4 |
polynomialis 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.
The monomial with the highest degree is 4xy^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.
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.
| Polynomial | Degree |
|---|---|
| - 5 x^2y + x^2y^4 - 11x^2 - 3 | 6 |
| x - 11x^4 +8x^3 | 4 |
| 7 | 0 |
| 0 | Undefined |
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.
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.
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.
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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 |
|---|---|
| 4x+x^(12) | It has a non-integer exponent. |
| 2x^2+5+y^(-2) | It has a negative exponent. |
| 3x/2 y+x^2-5y^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 |
|---|---|
| x/4-3x^3 | 2 |
| 2x^2+5-7y | 3 |
| xy/8+3x^3-3y^3+2x | 4 |
| Expression | Number of terms |
| 4x+x^(12) | 2 |
| 2x^2+5+y^(-2) | 3 |
| 3x/2y+x^2-5y^3+9 | 4 |
Discussion
Simplifying and Writing Polynomials in Standard Form
Concept
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.
a_n x^n + a_(n-1)x^(n-1) + ⋯ + a_1 x^1 + a_0x^0
In that general expression, n is a whole number and the coefficients a_n, a_(n-1), ..., a_2, a_1, a_0 are real numbers. The following expression written in standard form shows a polynomial with a degree of 5. x^5 -12x^4-2x^3+8x^2+9x+0 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 4. 3x^4 -2x^2+6x+5 ⇕ 3x^4+0* x^3-2x^2+6x+5
It can be seen that the x^3-term was omitted, at first. However, if chosen to do so, the polynomial can be expressed in standard form with a coefficient of 0 and the corresponding term.Concept
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). 7xy+2x-3xy-2x^2+x+5x^2 In this expression, there are three sets of like terms — namely, xy-terms, x-terms, and x^2-terms.
Method
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.
7xy+2x-3xy-2x^2+x+5x^2
There are three steps to follow to combine like terms.
1
Identify Like Terms
expand_more 
2
Grouping Like Terms
expand_more 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
Add and Subtract the Coefficients
expand_more 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.
7xy+2x-3xy-2x^2+x+5x^2 ⇕ 4xy+3x+3x^2
Example
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. &I. x^2+4x^3-5x^3+6x^2+2x^3 &II. 8x+4x^3-2x^5+7x-12x^3
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.
Answer
a Standard Form: x^3+7x^2
Classification: Binomial
Degree: 3
b Standard Form: - 2 x^5 -8x^3+15x
Classification: Trinomial
Degree: 5
Solution
a To simplify the polynomial, like terms will be identified and grouped.
x^2+6x^2 +4x^3-5x^3+2x^3
FactorOut
Factor out x^2 & x^3
(1+6)x^2+ (4-5+2)x^3
AddSubTerms
Add and subtract terms
7x^2+ x^3
| Term | Degree | Coefficient |
|---|---|---|
| 7x^2 | 2 | 7 |
| x^3 ⇔ ( 1)x^3 | 3 | 1 |
To write the polynomial in standard form, its terms will be arranged according to their degree, in descending order. Standard Form x^3+7x^2 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 ( 1)x^3, the polynomial's degree is 3 and its leading coefficient is 1. The following table summarizes all the information found.
| Standard Form | ||
|---|---|---|
| x^3+7x^2 | ||
| Classification | Degree | Leading Coefficient |
| Binomial | 3 | 1 |
b The same steps as in Part A will be followed. The polynomial will be simplified by identifying and combining like terms.
8x +4x^3-2x^5 +7x -12x^3
CommutativePropAdd
Commutative Property of Addition
8x+7x+ 4x^3-12x^3-2x^5
FactorOut
Factor out x & x^3
(8+7)x+ (4-12)x^3-2x^5
AddSubTerms
Add and subtract terms
15x -8x^3-2x^5
| Term | Degree | Coefficient |
|---|---|---|
| 15x ⇔ 15x^1 | 1 | 15 |
| - 8x^3 | 3 | - 8 |
| - 2x^5 | 5 | - 2 |
Next, the polynomial will be written in standard form sorting its terms by degree in descending order. Standard Form - 2 x^5 -8x^3+15x Finally, since the highest degree monomial is - 2 x^5, the polynomial's degree is 5 and its leading coefficient is - 2. The information found will be summarized using a table.
| Standard Form | ||
|---|---|---|
| - 2 x^5 -8x^3+15x | ||
| Classification | Degree | Leading Coefficient |
| Trinomial | 5 | - 2 |
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 | |
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