Question

What are all the possible rational zeros for `f(x)=2x^4-9x^2+7`?

Answer 1

The rational roots are `x = +-1`.

Explanation:

Let `a` represent `x^2`, thus `f(x)=2x^4-9x^2+7` can be rewritten as `2a^2 - 9a + 7`.

This is simply a matter of factoring now, remember, to find the zeroes, you must set the equation equal to zero:

`2a^2 - 9a + 7 = 0`

Using the quadratic formula:
(Note that a was used rather than x, as a is the variable used in the quadratic equation, we are finding the roots of a)

`a = (-b +- sqrt(b^2 - 4ac))/(2a)`
`= (-(-9) +- sqrt((-9)^2 - 4(2)(7)))/(2(2))`
`= (9 +- sqrt(81 - 56))/4`
`=(9+-sqrt(25))/4`
`=(9+-5)/4`

`:. a_1 = (9+5)/4 = 7/2`
`:. a_2 = (9-5)/4 = 1`

Recall `a = x^2`

Thus,
`:. x_1^2 = 7/2`
`:. x_1 = +- sqrt(7/2) = +- sqrt(14)/2`

`:. x_2^2 = 1`
`:. x_2 = +- 1`

Therefore, as the question is asking for rational roots only, `x = +- 1.`

Hope this helped, formatting took so long :P