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Excluded Values for Rational Expressions

Excluding values that result in division by zero

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Curiosity Rover
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Real World Applications – Algebra I


NASA’s Curiosity Rover Lands on Mars!

Student Exploration

On Monday, August 6, 2012, NASA’s Curiosity Rover landed on Mars, traveling a total of 8.5 months and 352 million miles. This project cost a whopping $2.5 billion to create!

One of the amazing abilities that this amazing vehicle can do is take pictures and send them back to earth for us to look at. Check out some of the pictures here: http://www.nasa.gov/mission_pages/msl/multimedia/gallery-indexEvents.html

A piece of metal on one of the mastcams of the rover has a surface area that’s represented by \begin{align*}x^2-x-2\end{align*} in square millimeters. If the width of the piece of metal is \begin{align*}x + 3\end{align*} milimeters, we can find and write the expression of the length of the piece of metal.

We can use the division of polynomials to find the length, and our knowledge that area is equal to the length multiplied by the width. Since we’re finding the length, we take the expression for area and divide it by the expression for the width.

\begin{align*}length=\frac{area}{width}= \frac{(x^2-x-2)}{(x+3)}\end{align*}

After reviewing the concept, we can use long division to divide these polynomials.

First I put the numerator inside the division sign, and our divisor on the outside. I now want to think about what \begin{align*}x\end{align*} can be multiplied by to get \begin{align*}x^2\end{align*}.

We know that \begin{align*}x \times x = x^2\end{align*}, so our first term is \begin{align*}x\end{align*}. Now, let’s multiply \begin{align*}x\end{align*} by \begin{align*}x + 3\end{align*}.

Now we want to subtract, and then bring down. Notice how I put parentheses around the whole expression, since we’re subtracting the whole expression.

Now I want to think about what can be multiplied by \begin{align*}x\end{align*} to give us \begin{align*}-4x\end{align*}.

Now that we’ve successfully divided these polynomials, we know that our answer is \begin{align*}x - 4\end{align*}.

When looking at numbers that \begin{align*}x\end{align*} can’t be, we’re looking at what would make the denominator zero, since we can’t divide by zero. Our original expression to represent the area is \begin{align*}\frac{(x^2-x-2)}{(x+3)}\end{align*}. Since we know that \begin{align*}-3 + 3 = 0\end{align*}, we can conclude that \begin{align*}x\end{align*} cannot equal -3.

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