How do you find all zeros of #f(x)=2x^4-2x^2-40#?

1 Answer
Feb 5, 2017

Answer:

#x = +-sqrt(5)" "# or #" "x = +-2i#

Explanation:

Given:

#f(x) = 2x^4-2x^2-40#

We can first treat this as a quadratic in #x^2# then factor the two resulting quadratic factors using the difference of squares identity:

#a^2-b^2 = (a-b)(a+b)#

as follows:

#f(x) = 2x^4-2x^2-40#

#color(white)(f(x)) = 2((x^2)^2-x^2-20)#

#color(white)(f(x)) = 2(x^2-5)(x^2+4)#

#color(white)(f(x)) = 2(x^2-(sqrt(5))^2)(x^2-(2i)^2)#

#color(white)(f(x)) = 2(x-sqrt(5))(x+sqrt(5))(x-2i)(x+2i)#

Hence zeros:

#x = +-sqrt(5)" "# or #" "x = +-2i#