Volume is length*width*height
V = LWH
Height = H
Width = H-2
Length = H+5
V = H(H-2)(H+5) = 30
H(H2+3H-10)=30
H3+3H2-10H = 30
H3+3H2-10H-30 = 0
Per the rational roots theorem the roots of the
polynomial will be within the factors of the
constant , -30, divided by the coefficient of
the highest term, 1 from H3
The real roots must be within the factors of -30/1
or the factors of -30
Factors of -30 are ± {1, 2, 3, 5, 6, 10, 15, 30}
We will use synthetic division to determine the roots.
I first tried 1, -1, 2, -2, 3, and determined that -3 is
a root of the polynomial H3+3H2-10H-30.
This means that (H+3) is a factor ... Note that if
H=-3 then H+3 = 0
The synthetic division looks like this:
-3 | 1 3 -10 -30 the root and the coefficients
| -3 0 30
---------------------------
1 0 -10 0
Since the last sum is 0, there is no remainder.
That means that H=-3 is a root of the polynomial.
The other terms from right to left become the
constant and coefficients of the new polynomial
in ascending order ... LEFT to RIGHT
H3+3H2-10H-30 = (H+3)(H2+0H-10)
(H+3)(H2-10) = 0
Since H=-3 will not work for positive height we
need to look at:
H2-10 = 0
H2 = 10
H = ±√10
Again, discarding the negative
Height = √10 inches
Width = H-2 = -2 + √10 inches
Length = H+5 = 5 + √10 inches
