To find all the zeros of f(x)=x4+6x2−7 means to find the values of x that make f(x)=0. In other words, it means finding solution of the equation x4+6x2−7=0 and for this we should factorize x4+6x2−7.
For this, let us split middle term in to two components 7x2 and −x2. Then x4+6x2−7=0 becomes
x4+7x2−x2−7=0 i.e. x2(x2+7)−1((x2+7)=0 or
(x2−1)(x2+7)=0. Note that (x2−1) can be further factorized into (x+1)(x−1). Hence, x4+6x2−7=0 can be written as
(x−1)(x+1)(x2+7)=0 and hence
Rational zeros of f(x) are {−1,1}.
Further if we include complex numbers in domain of x, from (x2+7)=0, we get (x−i√7)(x+i√7) or x={i√7,−i√7}
