Question

The remainder obtained when

x^4 + 3x^2 - 2x + 2 is divided by (x + b) is the square of the
remainder obtained when x^2 – 3 is divided by (x + b). Find the values of b.​

Answer 1

Use the polynomial remainder theorem: the remainder upon dividing a polynomial
`p(x)` by
`x-c` is
`p(c)`.

`(-b)^4+3(-b)^2-2(-b)+2=b^4+3b^2+2b+2`

`((-b)^2-3)^2=b^4-6b^2+9`

Now

`b^4+3b^2+2b+2=b^4-6b^2+9\implies9b^2+2b-7=0`

`\implies(9b-7)(b+1)=0`

`\implies b=\frac79\text{ or }b=-1`