Question

They ask me to measure the length, width, and height of a rectangular box in inches:
L: 6
W: 3
H: 2

Then it tells me to put it in v=lwh format and solve:
6*3*2 = 36

Then it says the following: Rewrite the formula using the variable x for the length. Substitute the value of the volume found in step 2 for V and express the width and height of the object in terms of x plus or minus a constant. For example, if the height measurement is 4 inches longer than the length, then the expression for the height will be (x + 4). Simplify the equation and write it in standard form. (continued)

Answer 1

continued: I didn't really understand what it was telling me to do but I tried and I wrote it like this: V = (x-3)(x-4) but I dont know if this is already a simplified form or if it is correct. when I write this equation in mathway.com or wolframalpha.com and tell it to simplify, it gives me V=x^2-7x+12
I dont know if this is correct or if that's the simplified version of V = (x-3)(x-4). please let me know.

If it IS correct, then I need to find the solutions to this equation algebraically using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.
I dont know how to solve using any of these! if you could explain (with the numbers that I am using) that would be awesome!

Answer 2

your equation is more or less correct
keep in mind that you're writing Length = x
and Width = x-3
and Height = x-4
volume V will be
\(\bf V = LWH \implies V = (x)(x-3)(x-4)\)

so yours is quite close, you just need to add the "x" for the volume
now the standard form for that, will be the same expression just expanded
so
\(\bf V = (x)(x-3)(x-4) \implies V = x(x^2-7x+12) \implies V = x^3-7x^2+12x\)

Answer 3

woops, darn got truncated :/

Answer 4

\(\bf \bf V = (x)(x-3)(x-4) \implies V = x(x^2-7x+12)\\ \implies V = x^3-7x^2+12x\)

Answer 5

and yes, you'd need the "fundamental theorem" and specially the "rational root test" and the others 2
but I'd say you just need to cover the material first
read the section on those theorems, if you need clearing out, then let us know :)

Answer 6

ok awesome thank you I understood everything you told me pretty well. so \[V=x^3-7x^2+12x\] is the "simplified" form of the previous equation I had (that you corrected me on)?

Answer 7

and is that how you write it in standard form?

Answer 8

yes, that'd be the "standard form" of the polynomial

Answer 9

how could I find the solution using fundamental theorem of algebra?

Answer 10

well, how many solutions will -> \(V=x^3-7x^2+12x\)
give you?

Answer 11

3

Answer 12

well, how did you know that?

Answer 13

becasue of the ^3 in x^3

Answer 14

ahhhh, well, THAT is what's called the \(\bf \color{blue}{\text{fundamental theorem of algebra}}\)

Answer 15

oh! ok! so how do I solve using rational root theorem?

Answer 16

http://www.youtube.com/watch?v=GTx-rXTQg1o