Finding zeroes of a function with multiplicities...

Question

      I have some other question on trying to understand this concept. I have an equation (x^3 + 4x^2 + 3x +12 = 0).

 

And the worksheet requests for me to find all of the zeros including multiplicities, but I don't know exactly how.

Answer 1

x³ + 4x² + 3x + 12 = 0

x²(x + 4) + 3(x + 4) = 0   factor by grouping

(x² + 3)(x + 4) = 0           common factor

 

Can you solve for x?

Answer 2

Looks like we can factor by grouping   (put in groups   x^3+4x^2        and      3x+12

 

x^2(x+4)       +3(x+4)

 

(x^2+3)(x+4) = 0

 

There are 3 roots all of multiplicities of 1

 

-4 is a root

 

x^2 = -3

 

x = +-sqrt(-3) =+- i*sqrt(3)

 

Our other 2 roots are:

 

i*sqrt(3)

 

-i*sqrt(3)

 

The last 2 roots are imaginary