Question

How do you find the zeros of a polynomial?

Answer 1

It depends...

Explanation:

If you are given the polynomial function in factored form, then each linear factor corresponds to a zero of the function and the number of times a given factor (or scalar multiple of it) occurs is the multiplicity of the corresponding zero.

For example:

`f(x) = 5(x+1)^3(x-2)`

has zeros `x=-1` with multiplicity `3` and `x=2` with multiplicity `1`.

The degree of the polynomial is the highest degree of any term when multiplied out. We can calculate this without multiplying out the factors by simply focusing on the terms of highest degree in each of the factors and adding them up.

For example, the degree of:

`(x^2+3)(x-1)^5" "` is `" "2 + 5*1 = 7`

If you can find all of the zeros of a polynomial with their multiplicities, then the degree of the polynomial is simply the count of the number of zeros taking into account their multiplicities.

In general it may not be easy or even possible to find algebraic expressions for the zeros of a polynomial function.

For example:

`f(x) = x^5+4x+2`

has one real zero and four complex zeros, but none of them are expressible using `n`th roots or other standard functions.

It is possible to find the zeros of any polynomial up to degree `4` algebraically, but it can get horribly messy.