consider the polynomial: p(x)=-3x^2(x-1)(x+1/3)(x+2)^3?

Question

1) what are the roots and multiplicities?

Answer 1

p(x)=-3x^2 (x-1) (x+1/3) (x+2)^3

1) what are the roots and multiplicities?
roots - multiplicity
0 .............2
1..............1
-1/3 .........1
-2 ............3
basically the multiplicity of a root A is the exponent of the factor (x-A)
.

Answer 2

To get the roots, put each factor equal to 0, e.g. one root comes from
x+1/3 = 0
therefore x = ... (you can do that one)

don't forget x is a factor too.

The multiplicity is just the power to which the factor is raised. Thus
-1/3 has multiplicity 1, so does one other root, another has multiplicity 2 and another has multiplicity 3. There, I haven't quite told you the answer, but don't need to, do I. -- not with Jim around!

Answer 3

The roots are just the factors. Since it's already factored for you, it's easy.

the x^2 means there are two roots of 0 (x - 0). The (x - 1) means there's one root of 1. (x + 1/3) means one root of -1/3, and (x+2)^3 means you have three roots of -2.

root: multiplicity
0: 2
1: 1
-1/3: 1
-2: 3