Find the polynominal with integers coefficients that satisfies the following conditions:

Question

Degree: 3 

Zeros: 2,-3i,3i

Constant coefficient: -108

 

A 6x^3-12x^2+54x-108

B x^3-2x^2+9x-108

C 6x^3-12x^2-54x-108

D 6x^3+12x^2+6x+108

E -x^3-2x^2+54x-108

F None of the above

Answer 1

The 3 roots yield 3 factors:

 

(x-2)(x-3i)(x+3i)

 

(x-2)(x²+9)

 

x³ - 2x² + 9x - 18  multiply by 6 to get a constant of 108

 

6x³ - 12x² + 54x - 108

Answer 2

The Constant coefficient being -108 means that the find number in the polynomial will be -108.  So that eliminates option D.

 

All zeros can be written as factors: (x - a) where a is the zero.  So your three zeroes could be written as the factors:

 

(x - 2), (x - (-3i)), and (x - 3i)  (note: (x-(-3i)) is really (x + 3i) because of the double negative).

 

You then multiply out the factors using FOIL (First Outside Inside Last).  It is best to focus on the complex numbers first, because they often simplify nicely.

 

(x + 3i)(x - 3i) = (x² -3ix +3ix -9i²) = x² + 9  (Remember: i² = -1)

 

We then FOIL the result with (x - 2)

 

(x - 2)(x² + 9) = x³ - 2x² + 9x - 18

 

This doesn't match any of the possible solutions, but you should also notice that the polynomial doesn't end in -108 either.  Since 6 • -18 = -108, we need to multiply the whole polynomial by 6.

 

6(x³ - 2x² + 9x -18) = 6x³ - 12x² +54x - 108

 

Therefore the solution is C.