Model and Solve Problems Involving Polynomial Functions

I try to draw cute, but that doesn't work out so well...

Good morning!

As for the reason why I'm up and about at 1:31 AM is beyond me, and I'm pretty sure I'm going to regret this later. So Mr. P, if I look like I'm going to pass out -- you will know why. Anyways, my name is Charina. For those of you who are still unaware of my apparent existence in your classroom, I'm the girl with the indigo travel mug with the nice-smelling tea and it's now my turn to teach all of you some math!

I had a hunch that I'll be doing the scribe, because the day before yesterday I randomly pulled out Word and started thinking about what I would say if I were to scribe. I wager it's because I'm on hiatus on my actual blog (yes, I blog) -- and the whole nostalgia that comes along with it.

Putting those aside, it's time to get serious. 

Yesterday, we finished what was left of the Polynomial Functions booklet which was "Model and Solve Problems Involving Polynomial Functions" which is, in layman's terms -- problem solving and graphing it. If you happened to be absent or didn't really understand what Mr. P was doing, then it would be wise for you to keep reading.

This sheet!

Should you have done Assignment #2 of this unit, figuring this out should be a little easier. If I recall well, there was a word problem on the second page where you have to increase the size of an object -- if you had that worksheet, then this problem is basically the opposite of that very question. This time, the situation calls to decrease the size:

Bill is preparing to make an ice sculpture. He has a block of ice that is 3ft wide, 4 ft high and 5ft long. Bill wants to reduce the size of the block of ice by removing the same amount from each of the three dimensions. He wants to reduce the volume of the ice block to 24ft^3.

a.) Write a polynomial function.

b.) How much should be removed in each dimension?

The first part wants us to write a polynomial function, but before that, we must figure out what numbers are we dealing with. Bill has a block of ice: 3ft wide, 4ft high and 5ft long. Does this remind you of something? It should. It's volume: Length x Width x Height! Since Bill wants to decrease the volume to 24ft^3, it will now be our new volume to solve for.

Our main goal is to find out what X is, however, to do that we must come up with the polynomial function. Using what we're given and what we are trying to accomplish, we'll use these facts write out our factored form:

We write out our factored form as ( 5- x ) ( 4 - x ) ( 3- x ) because it said in the problem that "Bill wants to reduce the size of the block of ice by removing the same amount from each of the three dimensions" therefore, we have to subtract an unknown ( x ) amount to each dimension. With 24 being our goal volume, this equation is telling us that (5 - unknown)(4 - unknown)(3 - unknown) will equal to 24.

I'll be honest and say that Mr. P's method was really confusing, however, I developed another way to solve this problem -- using the way I solved the word problem for Assignment #2. It works, so why not? 

First, we need to get the polynomial. How do we do it? We FOIL. After the FOIL-ing is finished, you get your polynomial equation. Hence, voila. 

Still remember Integral Zero? Good for you, because you'll need it in my method. What you have to do is to get the factors of the constant as shown above, and substitute X for a factor. Instead of looking for a zero, we're going to look for a factor that when it's plugged into the equation, it gives us 24. Yes, it's trial and error, but lucky for you because the answer is 1.

See?

Now that we know what X is, we also now know by how much each dimension should be decreased by to get it down to 24ft! Each dimension should be decreased by 1 in order to get to 24ft^3 of volume. However, that is not all -- we still have to graph it. But how do we graph it? Our x-intercepts! Be certain not to forget: ( 5- x ) ( 4 - x) ( 3- x ). Your y-intercept is determined by multiplying 5 x 3 x 4 = 60.

This is my sexy graph, which John criticised.
How dare you, you don't even go here.

Before I wrap up this post, I will give you the full list of the table that needs to be filled -- that I hope you guys know how to do for the test on Monday.

Degree: Cubic / X^3  (due to having three intercepts + it's on the equation)

Leading Coefficient: Negative (negative sign at the front of the equation)

End Behaviour: Quadrants 2 & 4

Zeros/ X-Intercepts: 5, 4, 3 (see above)

Y-intercept: 60 (see above)

Intervals: +ve = (infinity, 3] , [4, 5]

                      -ve = [3, 4] , [ 5, infinity)

I must rest now, before I turn into a walking zombie. See you!