Find the zeros and their multiplicity of the polynomial function f(x) = -3x2 (x-3) (x+4)3 (x2+ 5)

Question

this is pre calculus. need help understanding this.

Answer 1

f(x) = -3x²(x - 3)(x + 4)³(x² + 5)

 

If (x - a) if a factor of polynomial P(x), then a is a zero of P(x).

 

x = 0

x = 3

x = -4, three times

x = ±√5

Answer 2

The zeros (or roots) are the values of x that satisfy f(x) = 0.

 

f(x) = -3x² (x-3) (x+4)³ (x²+ 5)

 

Looking at the factors in turn, the roots are x=0 with a multiplicity of 2, x=3 with a multiplicity of 1, x=-4 with a multiplicity of 3.

 

The (x²+ 5) factor can be further factored as (x²+ 5) = (x-i√5)(x+i√5), giving two complex roots x= ±i√5, each with a multiplicity of 1.