# Polynomials Word Problems

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.

$$3395 – 11.60p$$

$$3395 – 11.60p$$

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:

$$x \cdot x = x^{2}$$

Blue:

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

White:

$$2y \cdot x = 2xy$$

Green:

$$x(x + 3y) = x^{2} + 3x$$

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.

$$2x^{2} + 7xy + 6y^{2}$$

$$2x^{2} + 7xy + 6y^{2}$$