How do you find all zeros of #f(x)=5x^4+15x^2+10#?

1 Answer
Dec 30, 2016

Answer:

#f(x)# has zeros #+-i# and #+-sqrt(2)i#

Explanation:

We will use the difference of squares identity, which can be written:

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

with #a=x# and #b=i# or #b=sqrt(2)i# as follows:

#f(x) = 5x^4+15x^2+10#

#color(white)(f(x)) = 5(x^4+3x^2+2)#

#color(white)(f(x)) = 5(x^2+1)(x^2+2)#

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

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

Hence the zeros of #f(x)# are:

#x = +-i#

#x = +-sqrt(2)i#