Question

What are all the possible rational zeros for `f(x)=x^4-7x^2+10` and how do you find all zeros?

Answer 1

`f(x)` only has irrational zeros: `+-sqrt(5)` and `+-sqrt(2)`

Explanation:

Given:

`f(x) = x^4-7x^2+10`

By the rational root theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `10` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1, +-2, +-5, +-10`

However, note that we can factor `f(x)` as a quadratic in `x^2` then using the difference of squares identity to find:

`x^4-7x^2+10 = (x^2-5)(x^2-2)`

`color(white)(x^4-7x^2+10) = (x^2-(sqrt(5))^2)(x^2-(sqrt(2))^2)`

`color(white)(x^4-7x^2+10) = (x-sqrt(5))(x+sqrt(5))(x-sqrt(2))(x+sqrt(2))`

So the only zeros of `f(x)` are irrational: `+-sqrt(5)`, `+-sqrt(2)`