Question

You have several boxes with the same dimensions. they have a combined volume of 4x4 − 21x3 − 46x2 + 219x + 180. determine whether the binomial below could represent the number of boxes you have.

Answer 1

The following are the choices for this problem:A. x+4 B. x+3 C. x-5 D. 4x+3

Factoring 4x^4-21x^3-46x^2+219x+180,  this will give us then the short answer of x+4.
The factor theorem states that if x-n is a factor of function f(x) = 4x^4-21^3-46^2+219x+180 then f(x) = 0. 
So let us evaluate f(-4) which gives 4(256) - 21(-64) -46(16) +219(-4) + 180 = 936 

Since the function f(-4) does not equate to zero, we could say that x+ 4 is not a factor. 
So using x+3: f(-3) = 4(81) - 21(-27) -46(9) +219(-3) + 180 = 0, so x+3 is a factor.