solving polynomial with degree 3...

**jonnygill** · March 11th 2011, 02:35 PM

the problem is...

$-x^3+12x^2-47x+24=0$

This polynomial was created for a word problem. The problem started off w/ this guy that had a block of ice 3x4x5=60 ft^3

He wants to reduce the volume TO (3/5) the original volume by removing the same # of feet from each dimension.

I made the equation... $(3-x)(4-x)(5-x)=V(x)$ to represent the volume of the cube in terms of the amount of feet removed. After multiplying out this problem becomes $V(x)=-x^3+12x^2-47x+60$

Since he wants to reduce the volume TO (3/5) the original volume, I can say that...

$(3/5)\cdot60=-x^3+12x^2-47x+60$

OR

$0=-x^3+12x^2-47x+24$

You can see this is the same polynomial that appears at the beginning of this post. I tried to find a whole number root (positive in this case only) and couldn't find one. I know the answer is ABOUT .6 feet. So, it makes sense that I can't plug in positive factors of 24 into the equation and get zero. Obviously, taking all the factors of 24 where one factor is a fraction is an absurd notion.

I can't use the quadratic formula for this problem either, since this polynomial is not a quadratic function. It has a constant, so it's not like I can remove x from the polynomial.

Any idea how I might solve this problem?

**topsquark** · March 11th 2011, 03:22 PM

Originally Posted by jonnygill

the problem is...

$-x^3+12x^2-47x+24=0$

You can solve it using Cardano's method, but I doubt that will give you any more insight on the answer. The only real solution is not rational...x = 0.597152101 or so. (Here's the exact solution.)

-Dan

**jonnygill** · March 11th 2011, 04:17 PM

Thanks Dan.

The book I am working from would not expect to me know or use Cardano's method. This section of the book is dealing with using synthetic division to solve polynomials. This problem occurs towards the end of the problem set, so it's one of the more difficult problems in this section. There is something I am missing, but I don't know what it is. I've looked this problem over for awhile now.

thanks.

**jonnygill** · March 11th 2011, 06:22 PM

Perhaps, instead of me asking how to solve a polynomial w/ a degree of 3 for x. I should ask this... given the word problem presented in the first post of this thread, how could i solve the problem. Keep in mind that the section of the book this problem is found in deals w/ using synthetic division to solve polynomial equations. And lets say that we are not allowed to use Cardano's method in this problem.

Thank you.

**Prove It** · March 11th 2011, 06:25 PM

Originally Posted by jonnygill

the problem is...

$-x^3+12x^2-47x+24=0$

This polynomial was created for a word problem. The problem started off w/ this guy that had a cube of ice 3x4x5=60 ft^3

He wants to reduce the volume by (3/5) by removing the same # of feet from each dimension.

I made the equation... $(3-x)(4-x)(5-x)=V(x)$ to represent the volume of the cube in terms of the amount of feet removed. After multiplying out this problem becomes $V(x)=-x^3+12x^2-47x+60$

Since he wants to reduce the volume by (3/5), I can say that...

$(3/5)\cdot60=-x^3+12x^2-47x+60$

OR

$0=-x^3+12x^2-47x+24$

You can see this is the same polynomial that appears at the beginning of this post. I tried to find a whole number root (positive in this case only) and couldn't find one. I know the answer is ABOUT .6 feet. So, it makes sense that I can't plug in positive factors of 24 into the equation and get zero. Obviously, taking all the factors of 24 where one factor is a fraction is an absurd notion.

I can't use the quadratic formula for this problem either, since this polynomial is not a quadratic function. It has a constant, so it's not like I can remove x from the polynomial.

Any idea how I might solve this problem?

It is possible that since you have not posted the entire question, what the question is actually asking is not being answered.

Please post the entire question...

Also, it is possible that since it says "reduce the amount by 3/5", he will only be left with 2/5 of the original volume...

**jonnygill** · March 11th 2011, 08:15 PM

Originally Posted by Prove It

It is possible that since you have not posted the entire question, what the question is actually asking is not being answered.

Please post the entire question...

Also, it is possible that since it says "reduce the amount by 3/5", he will only be left with 2/5 of the original volume...

I'm sorry, in the initial problem i said he has a cube of ice. He in fact has a three dimensional rectangular figure.

This problem is broken up into 4 parts: a, b, c, and d.

All parts concern a sculptor who has a block of ice that measures 3 x 4 x 5 = 60 ft^3. Before he starts sculpting, he wants to shave off the same number of feet from each dimension to make the block of ice smaller.

a asks to write a polynomial function to model the situation. I came up with $V(x)=-x^3+12x^2-47x+60$ This function shows Volume in terms of the amount removed from each dimension, x. The function itself was derived from (3-x)(4-x)(5-x) where x is the number of feet removed from each dimension.

b asks to graph the function (graphing calculator)

c states that he wants to reduce the volume TO 3/5 the original volume and to write an equation to model this situation... I figured that if $V(x)=60=-x^3+12x^2-47x+60$ then $\dfrac{3}{5}\cdot60=-x^3+12x^2-47x+60$ or $36=-x^3+12x^2-47x+60$ or $0=-x^3+12x^2-47x+24$

d asks how much he should remove from each dimension... essentially they are asking to solve $0=-x^3+12x^2-47x+24$ for x.

I tried solving the equation using synthetic division. I knew that x must be < 3 since the smallest dimension is 3 feet to begin with. I also figured that x must be > 0 because it is not practical to remove a negative length from a dimension. So, the only two factors of 24 that satisfy these requirements are 1 and 2 (i knew zero was not an option since there is a constant in the polynomial equation). I tried using synthetic division for both 1 and 2 and both resulted in a remainder that did not equal zero. And of course, I know the answer is about .6, so its not a surprise that using whole number factors was not fruitful.

How can I figure out how much to remove from each dimension so that the resulting volume is three fifths of the original volume without using Cardano's method?

Prove It: I made an error, the problem in fact said that the sculptor wants the new volume to be three fifths OF the original volume.

Thank you to whoever is kind enough to read through this thread.

**Prove It** · March 11th 2011, 11:27 PM

Originally Posted by jonnygill

I'm sorry, in the initial problem i said he has a cube of ice. He in fact has a three dimensional rectangular figure.

This problem is broken up into 4 parts: a, b, c, and d.

All parts concern a sculptor who has a block of ice that measures 3 x 4 x 5 = 60 ft^3. Before he starts sculpting, he wants to shave off the same number of feet from each dimension to make the block of ice smaller.

a asks to write a polynomial function to model the situation. I came up with $V(x)=-x^3+12x^2-37x+60$ This function shows Volume in terms of the amount removed from each dimension, x. The function itself was derived from (3-x)(4-x)(5-x) where x is the number of feet removed from each dimension.

b asks to graph the function (graphing calculator)

c states that he wants to reduce the volume TO 3/5 the original volume and to write an equation to model this situation... I figured that if $V(x)=60=-x^3+12x^2-37x+60$ then $\dfrac{3}{5}\cdot60=-x^3+12x^2-37x+60$ or $36=-x^3+12x^2-37x+60$ or $0=-x^3+12x^2-37x+24$

d asks how much he should remove from each dimension... essentially they are asking to solve $0=-x^3+12x^2-37x+24$ for x.

I tried solving the equation using synthetic division. I knew that x must be < 3 since the smallest dimension is 3 feet to begin with. I also figured that x must be > 0 because it is not practical to remove a negative length from a dimension. So, the only two factors of 24 that satisfy these requirements are 1 and 2 (i knew zero was not an option since there is a constant in the polynomial equation). I tried using synthetic division for both 1 and 2 and both resulted in a remainder that did not equal zero. And of course, I know the answer is about .6, so its not a surprise that using whole number factors was not fruitful.

How can I figure out how much to remove from each dimension so that the resulting volume is three fifths of the original volume without using Cardano's method?

Prove It: I made an error, the problem in fact said that the sculptor wants the new volume to be three fifths OF the original volume.

Thank you to whoever is kind enough to read through this thread.

I expect that since the factor theorem does not work, in that case you would have to use technology to solve the equation.

**jonnygill** · March 12th 2011, 09:39 AM

Thanks! Yeah, I accidentally wrote -37x instead of -47x. Prove It is correct, the authors of the book probably expected me to use technology (e.g. a graphing calculator) to determine the zeros of the function. I did this and I got x=.5971 or about .6

Thank you for your help. Somehow, it did not occur to me that in this case (i.e. this level of mathematics) using the graphing calculator is the way to go. Thanks again.

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