Question

How do you find all the zeros of `F (x) = 5x^3 (x+ 3)^ 4 (x-7)` with all its multiplicities?

Answer 1

The zeros are:

`x=0" "` with multiplicity `3`

`x=-3" "` with multiplicity `4`

`x=7" "` with multiplicity `1`

Explanation:

• Each linear factor corresponds to a zero. That is: `a` is a zero if and only if `(x-a)` is a factor.

• The multiplicity of each factor is the multiplicity of the corresponding zero.

We can rewrite the given `F(x)` slightly to bring it into a form with factors with the form `(x-a)` as follows:

`5x^3(x+3)^4(x-7) = 5(x-0)^3(x-(-3))^4(x-7)`

So the zeros are:

`x=0" "` with multiplicity `3`

`x=-3" "` with multiplicity `4`

`x=7" "` with multiplicity `1`