1. The volume of a box is V(x)=x^3-15x^2+66x-80. Find expressions for the dimensions of the box in terms of x.
2. The polynomial -x3-vx2+2x+w has a remainder of 4 when divided by x+2 and a remainder of 119 when divided by x-3. What are the values of v and w? (v & w eR)
After some tries of x = ±1, ±2 ...
I found V(2) = 8 - 60 + 132 - 80 = 0
so x-2 is a factor.
Using synthetic division, I got
x^3-15x^2+66x-80 = (x-2)(x^2 - 13x + 40)
= (x-2)(x-5)(x-8)
So a possible dimension for the box could be
(x-2) by (x-5) by (x-8)
2.
Let f(x) = -x3-vx2+2x+w
using the remainder theorem,
f(-2) = 8 -4v - 4 + w = 4 ---< 4v - w = 0
f(3) = -27 - 9v + 6 + w = 119 --> -9v + w = 140
add them
-5v = 140
v = -28
then -112 - w = 0
w = -112
check:
f(x) = -x^3 + 28x^2 + 2x - 112
f(-2) = 8 + 112 - 4 - 112 = 4
f(3) = -27 + 252 +6 -112 = 119
my answer is correct