Multiplicity??

Question

what multiplicities do the following equations have and why...please help

4x^3(x+2)(2x-1).... I though the 4x^3 would have a multiplicity of 3 because of the exponent but my teacher said its 1. Why?
x^3-8....and again why does this equation have a multiplicity of 1?

Answer 1

The multiplicity is the number of times a factor or a root appears.  x has a multiplicity of 3 because the exponent of x is 3.  So if we set the expression equal to zero, the root x=0 will have a multiplicity of 3.

 

The factors (x + 2) and (2x - 1) have multiplicities of 1 because their exponents are 1.  If set the expression equal to zero, the roots will be -2 and 1/2.  Both of these roots have a multiplicity of 1.

 

 

For more clarity, we can expand this out and set it equal to zero.

 

4 * x * x * x *(x + 2)(2x - 1) = 0

 

 

Now we set the factors equal to zero.

 

x = 0    ,   x = 0   ,   x = 0   ,     x + 2 = 0     ,       2x - 1 = 0

 

 

                                               x = -2        ,            x = 1/2

 

 

The root x=0 appears 3 times.  The other roots appear only once.