12.1: Polynomials
Introduction
A Visit Downtown
Mr. Travis is taking his Social Studies class on a tour of downtown. He has created a scavenger hunt for the students as they travel around the city. The scavenger hunt is made up of all different types of architecture and landmarks as well as problems that will need to be solved. Mr. Travis asked the bus to drop the students off in front of the town hall. In the square across from the town hall is a plaza with three cubes in it.
“Hey there is a problem with these cubes,” Tanya said to her friend Michael.
Here is the problem on the sheet.
Before you is a cube. Use the formula \begin{align*}A = 6s^2\end{align*} to find the surface area of a cube whose side measures 8 feet.
Tanya looked at Michael who looked at her. Both students began working on the problem in their notebooks.
You can work on this problem too. You will learn about polynomials in this lesson. By the end of the lesson you will be able to solve this problem.
What You Will Learn
In this lesson, you will learn the how to complete the following skills.
- Recognize and identify monomials, binomials and trinomials.
- Write and classify polynomials in standard form.
- Simplify polynomials by combining like terms.
- Evaluate polynomial expressions.
Teaching Time
I. Recognize and Identify Monomials, Binomials and Trinomials
Sometimes, you will see an expression or an equation that has exponents and variables in it. These expressions and equations can have more than one variable and sometimes more than one exponent in them. To understand how to work with these variables and exponents, we have to understand polynomials.
A polynomial is an algebraic expression that shows the sum of monomials.
Yes. They are new words. As we begin to work with polynomials, you will have to learn to work with brand new words.
Write each new word and its definition in your notebook.
A monomial is an expression in which variables and constants may stand alone or be multiplied. A monomial cannot have a variable in the denominator. We can think of a monomial as being one term.
To understand these new terms better, let’s look at some word prefixes so that we can better understand the new terms.
Word | Monoplane | Biplane | Triplane | Polygon |
---|---|---|---|---|
Definition | An airplane with one wing or one set of wings. | An airplane with two sets of wings | An airplane with three sets of wings | A shape with many sides. |
Prefix | Mono means one | Bi means two | Tri means three | Poly means many. |
Just like with airplanes, in math we can use these prefixes too. Each prefix will give us a hint as to the type of expression that we are dealing with.
Examples of monomials: \begin{align*}5 \quad x^3 \quad -2 x^5 \quad x^2y\end{align*}
Since the prefix mono means one, a monomial is a single piece or term. The prefix poly means many. So the word polynomial refers to one or more than one term in an expression. The relationship between these terms may be sums or difference.
Example of polynomials: \begin{align*}x^2+ 5 \qquad 3x-8+4x^5 \qquad -7a^2+9b-4b^3+6\end{align*}
We call an expression with a single term a monomial, an expression with two terms is a binomial, and an expression with three terms is a trinomial. An expression with more than three terms is named simply by its number of terms—“five-term polynomial.”
From the examples above, we can name the expressions as follows:
Number of Terms | 1 | 2 | 3 | 4 |
---|---|---|---|---|
Name | monomial | binomial | trinomial | four-term polynomial |
Expression | \begin{align*}-2x^5\end{align*} | \begin{align*}x^2+5\end{align*} | \begin{align*}3x-8+4x^5\end{align*} | \begin{align*}-7a^2+9b-4b^3+6\end{align*} |
Now that you understand how to identify each type of expression, let’s go and work on different ways to write them and classify them.
II. Write and Classify Polynomials in Standard Form
First, let’s think about how we can classify each polynomial. We classify them according to terms. Each term can be classified by its degree. The degree of a term is determined by the exponent of the variable or the sum of the exponents of the variables in that term.
Examples:
\begin{align*}x^2\end{align*} has an exponent of 2, so it is a term to the second degree.
\begin{align*}-2x^5\end{align*} has an exponent of 5, so it is a term to the fifth degree.
\begin{align*}x^2y\end{align*} has an exponent of 2 on the \begin{align*}x\end{align*} and an unwritten exponent of 1 on the \begin{align*}y\end{align*},
so this term is to the third degree \begin{align*}(2 + 1)\end{align*}. Notice that we add the two degrees together because it has two variables.
8 is a monomial that is a constant with no variable, its degree is zero.
We can also work on the ways that we write polynomials. One way to write a polynomial is in what we call standard form.
In order to write any polynomial in standard form, we look at the degree of each term. We then write each term in order of degree, from highest to lowest, left to write.
Example:
write the expression \begin{align*}3x-8+4x^5\end{align*} in standard form.
This is a trinomial. \begin{align*}3x\end{align*} has a degree of 1, -8 has a degree of zero, and \begin{align*}4x^5\end{align*} has a degree of 5. We write these in order by degree, highest to lowest:
\begin{align*}4x^5+3x-8\end{align*}
The degree of a polynomial is the same as the degree of the highest term, so this expression is called “a fifth-degree trinomial.”
III. Simplify Polynomials by Combining Like Terms
Begin this section by thinking about the following example.
Example
In a grocery store, a refrigerator in the back has 52 cartons of milk and 65 cans of soda. In the refrigerator near the cash registers, there are 12 cartons of milk and 26 cans of soda. How many do they have in all?
Yes...there are 64 cartons of milk and 91 cans of soda!
In this example, you probably added just the milk together and just the soda can together. You know that the milk cartons are alike. You know that soda cans are alike. But the milk and the soda cans are not alike. In mathematics, we are able to combine like terms but we do not combine unlike terms.
As we already saw, a term can be a single number like 7 or -5. These are called constants.
Any term with a variable has a numerical factor called the coefficient. The coefficient of \begin{align*}4x\end{align*} is 4. The coefficient of \begin{align*}-7a^2\end{align*} is -7. The coefficient of \begin{align*}y\end{align*} is 1 because its numerical factor is an unwritten. You could write “\begin{align*}1y\end{align*}” to show that the coefficient of \begin{align*}y\end{align*} is 1 but it is not necessary because any number multiplied by 1 is itself.
Terms are considered like terms if they have exactly the same variables with exactly the same exponents.
Examples:
\begin{align*}7n\end{align*} and \begin{align*}5n\end{align*} are like terms because they both have the variable \begin{align*}n\end{align*} with an exponent of 1.
\begin{align*}4n^2\end{align*} and \begin{align*}-3n\end{align*} are not like terms because, although they both have the variable \begin{align*}n\end{align*}, they do not have the same exponent
\begin{align*}5x^3\end{align*} and \begin{align*}8y^3\end{align*} are not like terms because, although they both have the same exponent, they do not have the same variable.
Like terms can be combined by adding their coefficients.
Examples:
\begin{align*}7n+5n &=12n\\ 3x^3+5x^3 &=8x^3\\ -2t^4-10t^4 &=-12t^4\\ 2n^2-3n+5n^2+11n &=7n^2+8n\end{align*}
Notice that the exponent does not change when you combine like terms. If you think of \begin{align*}7n\end{align*} as simply a shorter way of writing \begin{align*}n + n + n + n + n + n + n\end{align*} and \begin{align*}5n\end{align*} as a shorter way of writing \begin{align*}n + n + n + n + n\end{align*}, then combining those like terms to get \begin{align*}12n\end{align*} is a simpler way to write \begin{align*}7n + 5n\end{align*}.
IV. Evaluate Polynomial Expressions
In previous lessons, you have learned the order of operations, commonly called PEMDAS. In other words, arithmetic operations are performed in the following order:
- First any operations inside grouping symbols (P).
- Second any values with exponents (E).
- Third multiplication and division in order from left to right (M and D).
- Finally addition and subtraction in order from left to right (A and S).
When we consider expressions, we can evaluate an expression for a given value. In other words, we can find the total value if we know how much the variable is. We can replace the variable(s) with the given value and then use the order of operations to calculate the total value.
Example
Evaluate \begin{align*}x^2+3x-10\end{align*} for \begin{align*}x=5\end{align*}
Step 1: Replace the variables with the given value, 5.
\begin{align*}5^2+3 \cdot 5-10\end{align*}
Step 2: Find the total value using the order of operations.
\begin{align*} & 5^2+3 \cdot 5-10\\ & 25+3 \cdot 5-10 && (\text{There is no group, so first is the number with the exponent.})\\ & 25+15-10 && (\text{Complete the multiplication})\\ & 40-10 && (\text{Addition and subtraction from left to right.})\\ & 30 && (\text{Our total is} \ 30.)\end{align*}
This is our answer. We can evaluate any expression when we have been given a value for the variable.
Real-Life Example Completed
A Visit Downtown
Here is the problem from the introduction. Reread it and then solve it for the surface area of the cube.
Mr. Travis is taking his Social Studies class on a tour of downtown. He has created a scavenger hunt for the students as they travel around the city. The scavenger hunt is made up of all different types of architecture and landmarks as well as problems that will need to be solved. Mr. Travis asked the bus to drop the students off in front of the town hall. In the square across from the town hall is a plaza with three cubes in it.
“Hey there is a problem with these cubes,” Tanya said to her friend Michael.
Here is the problem on the sheet.
Before you is a cube. Use the formula \begin{align*}A =6s^2\end{align*} to find the surface area of a cube whose side measures 8 feet.
Tanya looked at Michael who looked at her. Both students began working on the problem in their notebooks.
Now solve for the surface area of the cube.
Solution to Real – Life Example
Now you can use the formula and the given information to solve for the surface area of the cube. The given length of the cube is 8 feet. You can substitute this into the formula for the side length.
\begin{align*}A &= 6s^2\\ A &=6(8^2)\\ A &=6(64)\\ A &=384 \ sq.feet \end{align*}
This is the surface area of the cube.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Polynomial
- an algebraic expression that shows the sum of monomials. A polynomial can also be named when there are more than three terms present.
- Monomial
- an expression where there is one term.
- Binomial
- an expression where there are two terms.
- Trinomial
- an expression where there are three terms.
- Constant
- a term that is a single number such as 4 or 9.
- Coefficient
- a variable and a numerical factor and the numerical factor is the coefficient
- Like Terms
- are terms that have the same variables and same exponents.
Time to Practice
Directions: Use the chart to identify each term with the correct label.
Number of Terms | 1 | 2 | 3 | 4 |
---|---|---|---|---|
Name | monomial | binomial | trinomial | four-term polynomial |
Expression | \begin{align*}-2x^5\end{align*} | \begin{align*}x^2+5\end{align*} | \begin{align*}3x-8+4x^5\end{align*} | \begin{align*}-7a^2+9b-4b^3+6\end{align*} |
- \begin{align*}4x^2\end{align*}
- \begin{align*}3x+7\end{align*}
- \begin{align*}9x^2+6y\end{align*}
- \begin{align*}x^2+2y^2+8\end{align*}
- \begin{align*}5c^3\end{align*}
- \begin{align*}3x^2+4x+3y^2+7\end{align*}
- \begin{align*}4x+3xy+9\end{align*}
- \begin{align*}2x^2+7y + 9\end{align*}
Directions: Determine the degree of each polynomial.
- \begin{align*}4x^2\end{align*}
- \begin{align*}5y^5\end{align*}
- 9
- \begin{align*}x^2+2y\end{align*}
- \begin{align*}7x^4+3y^3\end{align*}
- 12
Directions: Write the following polynomials in standard form and then identify its degree:
- \begin{align*}-8+3y^2-2y^3+y\end{align*}
- \begin{align*}m^4-12m^7+6m^5-6m-8\end{align*}
- \begin{align*}-x^3y^2+5x^3y+8xy\end{align*}
Directions: Simplify the following polynomials by combining like terms. Write your answer in standard form.
- \begin{align*}3x+7-5x+4\end{align*}
- \begin{align*}6y^2-4y^3+y^2-8\end{align*}
- \begin{align*}-5q+q^2+7-q-7\end{align*}
- \begin{align*}n^2m-3n^2m+5n^2 m^2+11n\end{align*}
Directions: Evaluate the following expressions for the given value.
- \begin{align*}7x^3\end{align*} for \begin{align*}x=2\end{align*}
- \begin{align*}6y^2-2y-8\end{align*} for \begin{align*}y = 6\end{align*}
- \begin{align*}3(x-7) + 5(x + 1)\end{align*} for \begin{align*}x = 10\end{align*}
- \begin{align*}-2y^3+6(y-4)^2+y\end{align*} for \begin{align*}y = -3\end{align*}