Find the value of a and b so that x⁴+x³+8x²+ax+b is divisible by (x²+1)

Let us devide the given polynomial x⁴+x³+8x²+ax+b by (x²+1) as shown above image:From the devision, we observe that the quotient is x²+x+7 and the remainder is (a-1)x+(b-7)since it is given that x⁴+x³+8x²+ax+b is exactly divisible by x²+1, therefore the remainder must be equal to 0 that is:(a-1)x+(b-7)=0=>(a-1)x+(b-7)=0 x+0=>(a-1)=0, (b-7) =0 (by comparing coefficients)=>a=1,b=7Hence, a=1 and b=7.

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