Writing a polynomial to represent a situation can help us answer questions and find solutions. Consider the following:

**Problem:**

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Sarina has published a book and wishes to send it to \(200\) readers. She has prepared \(200\) packages to mail. The cost of labor and materials to prepare for shipment is \($\,95.00\). The rates for shipping are:

\($16.50\)/ package international, \($4.90\)// package domestic (in the U.S.)Write a simplified polynomial to express the total cost to distribute her book if she ships \(p\) books within the U.S. and the rest internationally.

Think about what is known and unknown in the problem. Use the variable, \(p\), to represent the number of packages that Sarina will ship to domestic addresses.

\(p = domestic \,packages\)

Since the total number of packages is \(200\), and \(p\) are shipped domestically, the remaining books to be shipped outside the U.S. can be represented as \(200 – p\).

\(200 – p = international \,packages\)

There are \(3\) elements to the cost of distributing the books.

\(packaging = 95\)

The expression to represent shipping cost to each area comes from multiplying the cost per book by the number of books.

\(US \,shipping = 4.90p\)

\(Int’l\, shipping = 16.50(200 – p)\)

Add the three elements of the cost together to write an expression representing the total cost to distribute the books.

\(95 + 4.90p + 16.50(200 – p)\)

Use the distributive property.

\(95 + 4.90p +3300 – 16.50p\)

Combine like terms.

\(3395 – 11.60p\)

**Answer: **

Sarina can use this polynomial to find out the cost of shipping \(200\) books when \(p\) of them are shipped within the U.S. (domestically).

**Example**

**Problem**

A carpet designer creates a carpet that uses four colors according to the pattern and dimensions below. Express the area of the carpet as a polynomial.

Find the area of each colored piece of carpet by multiplying the length by the width.

**Red:**

**Blue: **

**White:**

**Green:**

Find the area of the whole carpet by combining the areas of the four pieces.

\(x^{2} + 2xy + 6y^{2} + 2xy + x^{2} + 3xy\)

Combine like terms.

\(2x^{2} + 7xy + 6y^{2}\)

**Answer: **

**Summary**

One of the powers of algebra is in representing aspects of the world with algebraic expressions in order to learn more about them. An expression that combines one or more terms to describe a situation is called a polynomial.

Binomials, which are polynomials with two terms, and monomials, which are polynomials with one term, are two types of polynomials. By definition, polynomials do not have variables in the denominator or negative exponents.