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Polynomials and Factoring

Count terms to identify polynomials

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Practice Polynomials and Factoring
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Recognize and Identify Monomials, Binomials and Trinomials

Have you ever tried to classify numbers? Take a look at this dilemma.

Sam saw this expression in his math book.

x28\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 Monopoly Bicycle Tricycle Polygon
Definition A situation in which one company owns all of the market for a given type of product. A vehicle with two wheels. A vehicle with three wheels. A shape with many sides.
Prefix Mono means one Bi means two Tri means three Poly means many.

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: 5x32x5x2y\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: x2+53x8+4x57a2+9b4b3+6\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 2x5\begin{align*}-2x^5\end{align*} x2+5\begin{align*}x^2+5\end{align*} 3x8+4x5\begin{align*}3x-8+4x^5\end{align*} 7a2+9b4b3+6\begin{align*}-7a^2+9b-4b^3+6\end{align*}

Identify each expression.

Example A

4x38\begin{align*}4x^3-8\end{align*}

Solution: Binomial

Example B

x2+3x+9\begin{align*}x^2+3x+9\end{align*}

Solution: Trinomial

Example C

6xy\begin{align*}6xy\end{align*}

Solution: Monomial

Now let's go back to the dilemma from the beginning of the Concept.

x28\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?

4x2x8y+4\begin{align*}4x^2x-8y+4\end{align*}

This expression has many terms. Therefore, it is called a polynomial.

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 2x5\begin{align*}-2x^5\end{align*} x2+5\begin{align*}x^2+5\end{align*} 3x8+4x5\begin{align*}3x-8+4x^5\end{align*} 7a2+9b4b3+6\begin{align*}-7a^2+9b-4b^3+6\end{align*}
1. 4x2\begin{align*}4x^2\end{align*}
2. 3x+7\begin{align*}3x+7\end{align*}
3. 9x2+6y\begin{align*}9x^2+6y\end{align*}
4. x2+2y2+8\begin{align*}x^2+2y^2+8\end{align*}
5. 5c3\begin{align*}5c^3\end{align*}
6. 3x2+4x+3y2+7\begin{align*}3x^2+4x+3y^2+7\end{align*}
7. \begin{align*}4x+3xy+9\end{align*}
8. \begin{align*}2x^2+7y + 9\end{align*}
9. \begin{align*}14xy\end{align*}
10. \begin{align*}4x^2+5x-9\end{align*}
11. \begin{align*}5x^3-4x^2+3x-10\end{align*}
12. \begin{align*}4x\end{align*}
13. \begin{align*}16x+4\end{align*}
14. \begin{align*}18x^2+5x-8\end{align*}
15. \begin{align*}9xyz\end{align*}
16. \begin{align*}5xy-6x\end{align*}
17. \begin{align*}18x^2-9x\end{align*}

Vocabulary Language: English

Binomial

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.
Monomial

Monomial

A monomial is an expression made up of only one term.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Trinomial

Trinomial

A trinomial is a mathematical expression with three terms.