Have you ever tried to classify numbers? Take a look at this dilemma.
Sam saw this expression in his math book.
\begin{align*}x^2-8\end{align*}
He isn't sure how to classify this expression. Do you know?
Expressions like this one are the focus of this Concept. Pay attention and you will know how to identify this expression by the end of the Concept.
Guidance
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.
Here are some 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.
Here are some 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 information 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*} |
Identify each expression.
Example A
\begin{align*}4x^3-8\end{align*}
Solution: Binomial
Example B
\begin{align*}x^2+3x+9\end{align*}
Solution: Trinomial
Example C
\begin{align*}6xy\end{align*}
Solution: Monomial
Now let's go back to the dilemma from the beginning of the Concept.
\begin{align*}x^2-8\end{align*}
This expression has two terms, therefore it is a binomial.
Vocabulary
- 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.
Guided Practice
Here is one for you to try on your own.
How would you identify the following expression?
\begin{align*}4x^2x-8y+4\end{align*}
This expression has many terms. Therefore, it is called a polynomial.
Video Review
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*}
- \begin{align*}14xy\end{align*}
- \begin{align*}4x^2+5x-9\end{align*}
- \begin{align*}5x^3-4x^2+3x-10\end{align*}
- \begin{align*}4x\end{align*}
- \begin{align*}16x+4\end{align*}
- \begin{align*}18x^2+5x-8\end{align*}
- \begin{align*}9xyz\end{align*}
- \begin{align*}5xy-6x\end{align*}
- \begin{align*}18x^2-9x\end{align*}