Question

How do you find the rest of the zeros given one of the zero `c=1/2` and the function `f(x)=2x^3-x^2-10x+5`?

Answer 1

`sqrt5` and `-sqrt5` are other two zeros of `f(x)=2x^3-x^2-10x+5`.

Explanation:

As `1/2` is a zero of function `f(x)=2x^3-x^2-10x+5`, `(x-1/2)` is a factor of `f(x)`. Note that coefficient of `x^3` is `2`, hence one can say `2(x-1/2)=2x-1` is a factor.

We can easily divide `f(x)=2x^3-x^2-10x+5` by `2x-1` as follows:

`f(x)=2x^3-x^2-10x+5`

= `x^2(2x-1)-5(2x-1)`

= `(2x-1)(x^2-5)`

It is easy to factorize `x^2-5` using `a^2-b^2=(a+b)(a-b)` and

`x^2-5=x^2-(sqrt5)^2=(x+sqrt5)(x-sqrt5)`

And hence `(x+sqrt5)` and `(x-sqrt5)` are also factors of `f(x)` and therefore

`sqrt5` and `-sqrt5` are other two zeros of `f(x)=2x^3-x^2-10x+5`.