Find a 3rd degree polynomial with roots 6 and -i with constant coefficient -24.

Question

A. f(x)=4x^3-24x^2+24x-24

B. f(x)=4x^3-24x^2+4x-24

C. f(x)=x^3-6x^2+x-24

D. f(x)=-4x^3+24x^2+4x-24

E. None of the above.

Answer 1

When we have complex roots, those roots have conjugates.   So our roots are

 

x₁ = 6

x₂ = -i

x₃ = i

 

 

Using these roots, we can create a function f(x).

 

f(x) =C(x - 6)(x + i)(x - i)

f(x) = C(x - 6)(x² + 1)

f(x) = C(x³ - 6x² + x - 6)

 

The constant coefficient is the last term and y-intercept of the function.  So evaluate C when f(0)=-24.

 

-24 = C(-6)

 

4 = C

 

 

Therefore,

 

f(x) = 4(x³ - 6x² + x - 6)

 

f(x) = 4x³ - 24x² + 4x - 24