How do you find all the zeros of # f(x) = x^4 + 6x^2 - 7#?

1 Answer
Mar 9, 2016

Answer:

Zeros of #f(x)# are #{-1,1,isqrt7,-isqrt7}#

Explanation:

To find all the zeros of #f(x)=x^4+6x^2-7# means to find the values of #x# that make #f(x)=0#. In other words, it means finding solution of the equation #x^4+6x^2-7=0# and for this we should factorize #x^4+6x^2-7#.

For this, let us split middle term in to two components #7x^2# and #-x^2#. Then #x^4+6x^2-7=0# becomes

#x^4+7x^2-x^2-7=0# i.e. #x^2(x^2+7)-1((x^2+7)=0# or

#(x^2-1)(x^2+7)=0#. Note that #(x^2-1)# can be further factorized into #(x+1)(x-1)#. Hence, #x^4+6x^2-7=0# can be written as

#(x-1)(x+1)(x^2+7)=0# and hence

Rational zeros of #f(x)# are #{-1,1}#.

Further if we include complex numbers in domain of #x#, from #(x^2+7)=0#, we get #(x-isqrt7)(x+isqrt7)# or #x={isqrt7,-isqrt7}#