If the product of the two roots of #x^4+px^3+qx^2+rx+s=0# is equal to the product of the other root then, a)#p^2s=r^2# b)#ps=r^3# What is the correct option?

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Answer 1 · 2017-10-30T04:52:12

(a) #p^2s=r^2#

Let the roots be #a,b,c# and #d#. Then

#(x-a)(x-b)(x-c)(x-d)=x^4+px^3+qx^2+rx+s=0#

and hence #a+b+c+d=-p#, #ab+ac+ad+bc+bd+cd=q#, #abc+abd+acd+bcd=-r# and #abcd=s#

Let #ab=cd#, then #abcd=a^2b^2=s# or #ab=sqrts#

and #-r=abc+abd+acd+bcd#

= #ab(c+d)+cd(a+b)#

= #ab(a+b+c+d)#

= #sqrtsxx-p# and squaring each side, we get

i.e. #r^2=sp^2# or #p^2s=r^2#

Hence, answer is (a).