if x2 +x+1 is a factor of x4 +ax+b then the value of 'a' and 'b' is

Given that:   x2 + x + 1 is a factor of   x4 + ax + b.

Since  x2 + x + 1  is a factor  x4 + ax + b

⇒  x4 + ax + b can be expressed as  –

 x4 + ax + b  =  (x2 + x + b) (x4 + k1x + k2)

⇒  x4 + 0x3 + 0x2 + ax + b  =  x4 + x3 + x2 + k1x3 + k1x2 + k1x + k2x2 + k2x + k2

⇒  x4 + 0x3 + 0x2 + ax + b  =  x4 + (k1 + 1) x3 + (1 + k1 + k2) x2 + (k1 + k2)x + k2

 Comparing the coefficients of x3 on both sides,  we get  – 

 k1 + 1  =  0  ⇒  k1=  – 1  ........  (1)

Again,  comparing coefficients of x2 on both sides,  we get – 

 0  =  1 + k1 + k2

   0  =  1 – 1 + k2

    k2  =  0  ........  (2)

Again comparing the coefficients of  x ,  we get –

  k1 + k2  =  a

   ⇒   a  =  –1 + 0

   ⇒   a  =  –1

and comparing constants on both sides,  we get –

         k2  =  b

   ⇒   b  =  k2  =  0

   ⇒   b  =  0

 Hence  ,  a  =  –1

   and       b =  0

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