Polynomials and Synthetic Division Word Problem

[Kale99]

The polynomial 2x^3 + 9x^2 + 4x - 15 represents the volume in cubic feet of a rectangular holding tank at a fish hatchery. The depth of the tank is (x-1) feet. The length is 13 feet.
A. Use synthetic division to help you factor the volume polynomial. How many linear factors should you look for? What are they?
B. Assume the length is the greatest dimension. Which linear factor represents the 13 ft length? What are the dimensions of the tank? What is it's volume? What is the value of x? Do you get the same volume if you substitute the volume of x into 2x^3 + 9x^2 + 4x -15?
C. Using a graphing calculator, analyze the graph. What is the relationship between the graph and the linear factors of the polynomial?

I feel entirely lost on this problem. I missed a few weeks of school because I had mono so I missed the polynomials chapter and am trying to play catch-up. How should I begin this problem? Should I divide the polynomial by x-1 with synthetic division? Sorry, I know I'm supposed to show my work but I'm not entirely sure how to even begin this problem.

 

[srmichael]

The polynomial 2x^3 + 9x^2 + 4x - 15 represents the volume in cubic feet of a rectangular holding tank at a fish hatchery. The depth of the tank is (x-1) feet. The length is 13 feet.
A. Use synthetic division to help you factor the volume polynomial. How many linear factors should you look for? What are they?
B. Assume the length is the greatest dimension. Which linear factor represents the 13 ft length? What are the dimensions of the tank? What is it's volume? What is the value of x?
C. Using a graphing calculator, analyze the graph. What is the relationship between the graph and the linear factors of the polynomial?

I feel entirely lost on this problem. I missed a few weeks of school because I had mono so I missed the polynomials chapter and am trying to play catch-up. How should I begin this problem? Should I divide the polynomial by x-1 with synthetic division? Sorry, I know I'm supposed to show my work but I'm not entirely sure how to even begin this problem.

Yes. Do synthetic division on x-1 and tell us what you get. Don't worry, we'll get you through this problem.

 

[Kale99]

Thank you . I divided the equation and I got 2, 11, 15, and 0 on the bottom. Would that mean 2x^2 + 11X + 15, Remainder 0?

 

[srmichael]

Thank you . I divided the equation and I got 2, 11, 15, and 0 on the bottom. Would that mean 2x^2 + 11X + 15, Remainder 0?

Yes! So now factor that quadratic and that answers the first part of part A. Can you do B and C?

 

[Kale99]

I factored that quadratic using the quadratic formula and I got -5/11, and -6/11 as answers once I reduced them. Is this correct? Are these the linear factors?

 

[srmichael]

I factored that quadratic using the quadratic formula and I got -5/11, and -6/11 as answers once I reduced them. Is this correct? Are these the linear factors?

That is incorrect, but since you did not show your work I do not know where you made your mistake.

This quadratic can actually be factored without the need for the quadratic formula, but since you used the quadratic formula I will show it here as well so you can see where you made your mistake.

$\displaystyle ax^2+bx+c=2x^2+11x+15$

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-11\pm\sqrt{11^2-4(2)(15)}}{2(2)}$

$\displaystyle x=\frac{-11\pm\sqrt{121-120}}{4}$

$\displaystyle x=\frac{-11\pm\sqrt{1}}{4}$

$\displaystyle x=\frac{-11\pm1}{4}$

$\displaystyle x=\frac{-11+1}{4} \ and\ x=\frac{-11-1}{4}$

$\displaystyle x=\frac{-10}{4} \ and\ x=\frac{-12}{4}$

$\displaystyle x=\frac{-5}{2} \ and\ x=-3$

If you now take each of these values and setr them equal to 0 you will have the two linear factors: (2x+5) and (x+3). So the polynomial factors like so:

$\displaystyle 2x^2+11x+15=(x-1)(2x+5)(x+3)$

 

[JeffM]

Thank you . I divided the equation and I got 2, 11, 15, and 0 on the bottom. Would that mean 2x^2 + 11X + 15, Remainder 0? YES indeed

Let's try a more roundabout approach.

The formula for volume of a rectangular prism (a rectangle in three dimensions) is length * width * depth. So it is logical that an expression for such a volume would be a polynomial of degree 3, a cubic. You probably got that on your own, but it never hurts to make sure.

They tell you that the depth is (x - 1) feet so if you divide (x - 1) into the the polynomial, you will get the product of the length and width, a quadratic that tell you the area of the floor. This explains WHY you are dividing by (x - 1).

$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ 2x^2 + 11x + 15$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$
$\displaystyle x - 1\ |2x^3 + 9x^2 +4x - 15$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ 2x^3 - 2x^2$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \_\_\_\_\_\_\_\_$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11x^2 + \ \ 4x$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11x^2 - 11x$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \_\_\_\_\_\_\_\_\_$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15x - 15$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15x - 15$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \_\_\_\_\_\_\_\_$
$\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0$

Now you factor $\displaystyle 2x^2 + 11x + 15 = (2x + 5)(x + 3).$

You are NOT looking for the roots of this polynomial.

One of those factors represents the length and the other the width. Which is the length, given that you are told that the length is the greatest dimension?

So what is x?

So what is the width?

So what is the depth?

So what is the volume?

 

[Kale99]

Would that mean 2x+5 is the length and x+3 is the width?

EDIT: For Part A, my mistake was that I substituted 2B into the bottom half of the equation and not 2A. Thanks for clearing that up

 

[srmichael]

Would that mean 2x+5 is the length and x+3 is the width? CORRECT

EDIT: For Part A, my mistake was that I substituted 2B into the bottom half of the equation and not 2A. Thanks for clearing that up

So can you tackle Part B and C now?

 

[JeffM]

Would that mean 2x+5 is the length and x+3 is the width? That is correct, but I fear you are guessing, which is why you asked for confirmation.

EDIT: For Part A, my mistake was that I substituted 2B into the bottom half of the equation and not 2A. Thanks for clearing that up

If 2x + 5 is the width, then, because the tank is longer than it is wide, 2x + 5 < x + 3, correct?

$\displaystyle 2x + 5 < x + 3 \implies 2x - x < 3 - 5 \implies x < - 2.$

So the depth = x - 1 < - 3, which is impossible.

So now answer the questions in my previous post, and you have all the information that you need.

 

[Kale99]

Okay so for Part B, since 2x + 5 is the length then 2x+ 5 = 13, correct? This would mean x=4. This would also mean the width (x+3) = 7. The depth is x-1 which = 3. The volume would be the three numbers multiplied together which would equal 273. Is this correct?

 

[Kale99]

Assuming that is correct, I'm a bit lost on Step C. What am I supposed to graph?

 

[JeffM]

Okay so for Part B, since 2x + 5 is the length then 2x+ 5 = 13, correct? This would mean x=4. This would also mean the width (x+3) = 7. The depth is x-1 which = 3. The volume would be the three numbers multiplied together which would equal 273. Is this correct?

It is. But you still have not grasped that the cubic equalled the volume. You should learn to check your work, a habit that will give you confidence and save your bacon on tests.

$\displaystyle 2x^3 + 9x^2 + 4x - 15 = 2 * 4^3 + 9 * 4^2 + 4 * 4 - 15 = 2 * 64 + 9 * 16 + 16 - 15 = 128 + 160 - 15 =128 + 145 = 273.$

Edit: We cross-posted. You are supposed to graph the polynomial and explain, given the context of the problem, what the graph tells you.

 

[Kale99]

You're right. I think I misread the problem by reading it over so many times. Since this is about a rectangular prism, the volume is length * width * depth so it would obviously be a cubic polynomial. I'm not sure how I missed that. I definitely need to work on checking my work in the future.

 

[Kale99]

Okay so I graphed the polynomial 2x^3+9x^2+4x-15 and I'm trying to find the relationship between that and the linear factors of x=-3, and x=-5/2. I noticed that the graph crosses the x axis at x=-3 and x=1. Does that have to do with the relationship?

 

[JeffM]

Okay so I graphed the polynomial 2x^3+9x^2+4x-15 and I'm trying to find the relationship between that and the linear factors of x=-3, and x=-5/2. I noticed that the graph crosses the x axis at x=-3 and x=1. Does that have to do with the relationship?

The linear factors are NOT x = - 3 and = -5/2. Those are two of the three zeroes or roots of the cubic. The linear factors are (x - 1),
(2x + 5), and (x + 3). (x - 1) is the depth, (2x + 5) = the length, and (x + 3) = the width. So what does their product represent? And what type of number must it be to have practical meaning in the context of this problem. So the graph is a graph of what, subject to what constraints, in the context of this problem?

You need to understand that algebra is a general but abstract tool that can be applied to trillions of practical problems. What things mean is determined by the specific problem that is being addressed. Part of what this question is asking you to do is to think about what these expressions mean in a specific case.

 

[Kale99]

You're right. My bad. I'm at the point of making careless mistakes since I've been trying to do so much to catch up. I really do need to work on checking my work. The product represents the volume of the fish tank. I'm not sure what type of number it needs to be other than positive of course. I'm not even sure what I'm graphing at this point. I know the polynomial represents the volume but otherwise I feel pretty lost on this problem.

 

[JeffM]

You're right. My bad. I'm at the point of making careless mistakes since I've been trying to do so much to catch up. I really do need to work on checking my work. The product represents the volume of the fish tank. I'm not sure what type of number it needs to be other than positive of course. I'm not even sure what I'm graphing at this point. I know the polynomial represents the volume but otherwise I feel pretty lost on this problem.

Exactly. The cubic represents the volume of a fish tank and so must be a positive number. The only part of the graph that is relevant in this context is that part where x > 1 so that the cubic is positive and every length is positive.

$\displaystyle y = 2x^3 + 9x^2 + 4x - 15 = volume\ of\ a\ certain\ class\ of\ rectangular\ prisms.$

But what class of rectangular prism?

The answer is that class where the depth is four units less than the width and the length equals 5 units plus 2 times the sum of the depth plus one unit. The graph gives you the volume of every rectangular prism with those relationships among length, width, and depth, an infinite amount of information stuffed into one graph. This is the economic aspect of mathematical thinking.

 

[mrsir]

so

Exactly. The cubic represents the volume of a fish tank and so must be a positive number. The only part of the graph that is relevant in this context is that part where x > 1 so that the cubic is positive and every length is positive.

$\displaystyle y = 2x^3 + 9x^2 + 4x - 15 = volume\ of\ a\ certain\ class\ of\ rectangular\ prisms.$

But what class of rectangular prism?

The answer is that class where the depth is four units less than the width and the length equals 5 units plus 2 times the sum of the depth plus one unit. The graph gives you the volume of every rectangular prism with those relationships among length, width, and depth, an infinite amount of information stuffed into one graph. This is the economic aspect of mathematical thinking.

so what is the answer to part C?

 

[lookagain]

That is incorrect,
but since you did not show your work I do not know where you made your mistake.

This quadratic can actually be factored without the need for the quadratic formula,
but since you used the quadratic formula I will show it here as well so you can see
where you made your mistake.

ax^2+bx+c= > > 2x^2+11x+15 < <This is a quadratic polynomial.


2x^2 + 11x + 15 = 0 is the related quadratic equation.


$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-11\pm\sqrt{11^2-4(2)(15)}}{2(2)}$

$\displaystyle x=\frac{-11\pm\sqrt{121-120}}{4}$

$\displaystyle x=\frac{-11\pm\sqrt{1}}{4}$

$\displaystyle x=\frac{-11\pm1}{4}$

$\displaystyle x=\frac{-11+1}{4} \ and\ x=\frac{-11-1}{4}$No, the connecting word is to be "or." **

$\displaystyle x=\frac{-10}{4} \ and\ x=\frac{-12}{4}$Ditto.

$\displaystyle x=\frac{-5}{2} \ and\ x=-3$Ditto.

If you now take each of these values and [set] them equal to 0
No, you don't set the values equal to zero. You set the factors equal to zero.


you will have the two linear factors: (2x+5) and (x+3). So the polynomial factors like so:

$\displaystyle 2x^2+11x+15=(x-1)(2x+5)(x+3)$No, that quadratic is not equal to that cubic.

Instead, 2x^3 + 9x^2 + 4x - 15 = (x - 1)(2x + 5)(x + 3).

** Both factors don't have to be equal to 0 at the same time for their product (the quadratic) to equal 0.

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