Question

If the polynomials (2x^3+ax^2+3x-5)and (x^3+x^2-2x+a) leave the same remainder when divided by (x-2). Find the value of a. Also find the remainder in each case

Answer 1

The polynomial remainer theorem says that a polynomial
`p(x)` leaves a remainder of
`p(c)` when divided by
`x-c`. When
`x=2`, we have

`2x^3+ax^2+3x-5=4a+17`

`x^3+x^2-2x+a=a+8`

Both operations leave the same remainder, which means

`4a+17=a+8\implies3a=-9\implies\boxed{a=-3}`

Then the remainders are both 5.