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Polynomial Word Problems

As of right now, you should know how to add, subtract, and multiply polynomials.

So when we look at word problems they'll only be asking you to add, subtract, or multiply polynomials.

The trickier part is figuring out which they want you to use. Fortunately, the problems usually deal with shapes which can help clue us in on what to use. If it references perimeter, chances are you need to use addition or subtraction. If references area or volume, chances are you need to use multiplication.

Let's look at an example.

Linda has a rectangular garden that's $3x-5$ feet by $2x+8$ feet. She wants to put a fence around the perimeter of the garden. If each foot of fence costs 5 dollars, how much will the total fencing around the garden cost?

Step 1: Draw a Diagram

Seriously, don't skip this step. Diagrams are a great way to help visualize the problem and keep things in order. Plus you get to make a quick doodle.

Here's Linda's garden. Each side is either $3x-5$ feet or $2x+8$ feet.

Step 2: Figure out What to Use

We want to find the price of the fencing that will surround the garden. To do this, we first need to figure out how much fencing there is.

This is a perimeter problem. Which means, we'll need to add up the sides.

Step 3: Set up the Problem

The last trickier step, is to set the problem up. First, we need to find the perimeter of the rectangle. $$\text{perimeter}=(3x-5)+(2x+8)+(3x-5)+(2x+8)$$ Then, we multiply this by the price per foot of fencing, which is $5$ dollars. $$5(\text{perimeter})=5[(3x-5)+(2x+8)+(3x-5)+(2x+8)]$$

Step 4: Solve

In my opinion, this is the easiest step of word problems. We already have it all set up, and finally we just add together some polynomials which we already know how to do. $$\begin{align} &5[(3x-5)+(2x+8)+(3x-5)+(2x+8)]\\ &=5[3x-5+2x+8+3x-5+2x+8]\\ &=5[(3x+2x+3x+2x)+(-5+8-5+8)]\\ &=5[10x+6]\\ &=50x+30 \end{align}$$ It will cost $50x+30$ dollars to put fencing around this garden.

Polynomial Word Problems

Word problems involving binomials are often related to basic geometry (like the concept of area, for example).

An important thing to remember is that word problems are just presenting you with familiar information in a new way (i.e., in words). You already know how to handle these problems, so don't let the fact that they're word problems throw you off.

Let's look at an example:

First off, we know that we can find the area of a rectangle by multiplying its length by its width.

Note that we're not solving for an actual number here—the problem directs us to find the area of the rectangle "in terms of $x$." This signals to us that we'll have the variable $x$ in our final answer.

Now, let's apply the formula we previously recalled to the rectangle above:

Notice how we used FOIL here, just like we've done with other binomials. Remember, FOIL works whenever you're multiplying two binomials together, whether they're in a word problem or not.

(Video) Polynomials 06 Multiply Polynomials Word Problem

by Mister Zuidema

This video by Mister Zuidema walks through a word problem with polynomials.

$\large{\text{Summary}}$

Let's look at it.

Sally has a photo which is $20\text{ cm}$ tall and $32\text{ cm}$ wide. She wants to put a frame around the photo which is the same width all around. What is the expression for the area of the framed photo.

First, we draw out the diagram of the photo and the frame. We don't know the width of the frame, so we label it $x$. We want the area of the whole frame, so we need to write an expression that is equal to the length times the height of the frame. We know the photo is $32\text{ cm}$ wide and the frame is $x$ wide on each side. So, the length of the frame is $(32+2x)$. We find the height of the frame similarly. We know the height of the photo is $20$ and each side of the frame is $x$, so the height is $(20+2x)$. Now, the area is the length times the width, so we multiply binomials. In the video, he uses a different way of multiplying, but I will be using FOIL. We see,

$$\begin{align} (32+2x)\times(20+2x)&=(32)(20)+(32)(2x)+(2x)(20)+(2x)(2x)\\ &=640+64x+40x+4x^2\\ &=640+104x+4x^2 \end{align}$$

The expression that represents the area of the whole frame is $640+104x+4x^2$ which is equal to $$4x^2+104x+640$$