If #a# and #b# are integers such that #x^2 -x-1# is a factor of #ax^3 + bx^2 +1#, then find b?

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Answer 1 · 2017-06-07T15:20:14

#b=-2#

If #x^2-x-1# is a factor of #a x^3 + b x^2 + 1# then

#a x^3 + b x^2 + 1=(c x + d) (x^2 - x - 1)# then grouping coefficients

#(a-c)x^3+(b+c-d)x^2+(c+d)x+d+1 = 0#

This must be true for all #x# so solving

#{(d+1=0),(c+d=0),(b+c-d=0),(a-c=0):}#

we have

#a=1,b=-2,c=1,d=-1# so

#b=-2#