Finding Real Roots of Polynomial Equations!

Question

The width of a container is 5 feet less than its height. Its length is 1 foot longer than its height. The volume of the container is 216 cubic feet. How tall is the container?

 

A. 3ft

B. 8ft

C. 4ft

D. 9ft

Answer 1

h represents the height

h - 5 represents the width

h + 1 represents the length

 

h(h - 5)(h + 1) = 216          FOIL

h(h² - 4h - 5) = 216            distribute

h³ - 4h² - 5h = 216             subtract 216 from both sides

h³ - 4h² - 5h - 216 = 0        

 

By the rational root theorem, the possible roots are:

±1, ±2, ±3, ±4, ±6, ±8, ......

The first four contradict the conditions.

 

6 is too small for the volume of 216

8 is the solution.

 

B. 

 

 

Answer 2

V=l*w*h

V=(h+1)(h-5)(h)

V=h^3-4h^2-5h after multiplying

216=h^3-4h^2-5h

h^3-4h^2-5h-216=0

roots of the equation are factors of 216

216=2*2*2*3*3*3

notice that the length and height are consecutive integers (h and h+1)

in 2*2*2*3*3*3, you have (2*2*2) and (3*3) which are consecutive integers (8 and 9)

h=8 ft, l=9 ft, and w=3 ft

if you did not notice this then use factors of 216 to find the roots

factors of 216 would be 2,-2,4,-4,3,-3,9,-9,8,-8,12,-12, and so on

you can get all of them from the factorization of 216

however, if you try 8 you will see that it is a root

8^3-4*8^2-5*8=216

512-256-40=216

512-296=216

216=216

so 8 is a root and 8 fits the descriptions of the width and length

8+1=9 and 8-5=3 and 9*8*3=216

B. 8 ft is the answer