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

1 Answer
Nov 5, 2016

Answer:

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