Solving Polynomal Equations

  • #1
1. The height,length and width of a small box are consecutive intergers with the height being the smallest of the three dimensions. If the length and width are increased by 1 cm each and the height is doubled, then the volume is increased by 120 cm^3. Find the dimensions of he original small box



2. v=lwh



3. I tried to appoach this two different ways but couldn't get either to work:

1. h = x
w = x+1
l= x+2
therefore (2x)(x+2)(x+3) = 120
2x^3 + 10x^2+12x-120 = 0
i tried to use the factor theroem to solve the equation but could not get it to work.
p{+-1,+-2,+-3...}
q{+-1,+-2}
p/q {+-1,+-1/2, +-2, +-3, +-3/2...}

I knowthat in order to solve the question, I am going to need to need to find the roots of the polynomial and therefore have to solve for x. The factor theroem indicates that if i plug in p/q into the above given equation, it will equal zero, and i can finish the question. The only problem i am having is determining which p/q value satisfies the equation. Another solution would be to factor, but i tried that and it ended up being..

2x^2(x+5) + 12(x-10) and since there are no more common factors, it cannot be taken any further. As a result, i am pretty sure i have to use the factor theorem, but i can't figure out what p/q value satisfies the equation

So then i tried (2x)(x+2)(x+3)= 120 + (x)(x+1)(x+2)
and got..
x^3+7x^2-10x-120 = 0
and tried to use factor theorem again but couldn't get it to work.


Thanks in advance for the help.
 
Last edited:

Answers and Replies

  • #2
312
0
That second equation is what you need to solve. You made some sort of algebra mistake when you expanded, the answer is a nice integer
 
  • #3
cristo
Staff Emeritus
Science Advisor
8,107
73
3. I tried to appoach this two different ways but couldn't get either to work:

1. h = x
w = x+1
l= x+2
therefore (2x)(x+2)(x+3) = 120
2x^3 + 10x^2+12x-120 = 0
i tried to use the factor theroem to solve the equation but could not get it to work.
p{+-1,+-2,+-3...}
q{+-1,+-2}

So then i tried (2x)(x+2)(x+3)= 120 + (x)(x+1)(x+2)
and got..
x^3+7x^2-10x-120 = 0
and tried to use factor theorem again but couldn't get it to work.


Thanks in advance for the help.
Your second method is correct. You have made a mistake in your last line though; it should read x^3+7x^2+10x-120 = 0.

Try and find a factor here (it shouldn't take too many tries!) Note that you do not need to try negative values, since x is a length, and a negative length is not physically valid.
 
  • #4
ok i see it now, can't believe i had negative 10x..thanks for the help
 
  • #5
AlephZero
Science Advisor
Homework Helper
6,994
292
So then i tried (2x)(x+2)(x+3)= 120 + (x)(x+1)(x+2)
That's correct.

and got..
x^3+7x^2-10x-120 = 0
Check your algebra. The "-10x" is wrong.
 

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