Question

A cuboid has volume equal to the polynomial, $p\left(x\right)=x^3+12x^2+47x+60$. It is given that $p(-3)=p(-4)=p(-5)=0$. Choose the possible sides of the cuboid.

• A. $(x+3),(x+4),(x+5)$
• B. $(x-3),(x-4),(x-5)$
• C. $1,1,(x+3)(x+4)(x+5)$
• D. $-1,1,(x+3)(x+4)(x+5)$

Answer 1

Answer: (A), (C)

The correct options are
A $(x+3),(x+4),(x+5)$
C $1,1,(x+3)(x+4)(x+5)$

Let the dimensions of the cuboid be given by $l,b\&h$.
Then the volume of the cuboid is $V=lbh$.
Clearly each of the dimension is a factor of its volume.

According to factor theorem, (x-a) is a factor of polynomial p(x) if p(a) = 0.
It is given that p(-3) = p(-4) = p(-5) = 0
This means that (x+3), (x+4) and (x+5) are the factors of the polynomial. Hence, combinations of these represent the dimensions of the cuboid.