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

2 Answers
Jul 9, 2016

Answer:

#x = +-(sqrt((2sqrt(10)+1)/20)) +- (sqrt((2sqrt(10)-1)/20))i#

Explanation:

Since #f(x)# has no odd terms, we could treat it as a quadratic in #x^2#, but the resulting quadratic has negative discriminant:

#Delta = (-1)^2-4(5)(2) = 1-40 = -39#

As a result it has only Complex zeros, of which we would then want to find square roots. This is a possible approach, but there is an alternative one...

Note that:

#(a^2-kab+b^2)(a^2+kab+b^2) = a^4 + (2-k^2)a^2b^2 + b^4#

Let:

#a = root(4)(5)x#

#b = root(4)(2)#

Then:

#a^4 + (2-k^2)a^2b^2 + b^4=5x^4+(2-k^2)sqrt(10)x^2+2#

If we let:

#k = sqrt((20+sqrt(10))/10)#

Then:

#(2-k^2)sqrt(10) = (2-(20+sqrt(10))/10)sqrt(10) = -1#

And:

#kab = sqrt((20+sqrt(10))/10)root(4)(10)x = sqrt((20+sqrt(10))/10*sqrt(10)) = sqrt(2sqrt(10)+1)#

So:

#f(x) = (a^2-kab+b^2)(a^2+kab+b^2)#

#=(sqrt(5)x^2-(sqrt(2sqrt(10)+1))x+sqrt(2))(sqrt(5)x^2+(sqrt(2sqrt(10)+1))x+sqrt(2))#

Then the zeros of these quadratic factors are given by the quadratic formula as:

#x = (+-sqrt(2sqrt(10)+1)+-sqrt((2sqrt(10)+1)-4sqrt(10)))/(2sqrt(5))#

#=+-(sqrt((2sqrt(10)+1)/20)) +- (sqrt((2sqrt(10)-1)/20))i#

Jul 9, 2016

Answer:

#x=+-(sqrt((2sqrt(10)+1)/20)) +- (sqrt((2sqrt(10)-1)/20))i#

Explanation:

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

I will use the result I derived for another question (see https://socratic.org/s/aw38evei), that the square roots of #a+-bi# are:

#+-(sqrt((sqrt(a^2+b^2)+a)/2)) +- (sqrt((sqrt(a^2+b^2)-a)/2))i#

First treat #f(x)# as a quadratic in #x^2# and use the quadratic formula to find:

#x^2=(1+-sqrt((-1)^2-4(5)(2)))/(2*5) = (1+-sqrt(-39))/10 = 1/10+-sqrt(39)/10i#

Let #a=1/10# and #b=sqrt(39)/10#

Then:

#sqrt(a^2+b^2) = sqrt(1/100+39/100) = sqrt(40/100) = (2sqrt(10))/10#

So the square roots of #a+bi# are:

#+-(sqrt((sqrt(a^2+b^2)+a)/2)) +- (sqrt((sqrt(a^2+b^2)-a)/2))i#

#=+-(sqrt(((2sqrt(10))/10+1/10)/2)) +- (sqrt(((2sqrt(10))/10-1/10)/2))i#

#=+-(sqrt((2sqrt(10)+1)/20)) +- (sqrt((2sqrt(10)-1)/20))i#

These are the zeros of our quartic.