Question

When x³ − 2x² + ax − b is divided by x² − 2x − 3, the remainder is x − 6. The values of a and b are respectively

Options

• −2, −6

•  2 and −6

• - 2 and 6

•  2 and 6

Answer 1

If the reminder (x −6) is subtracted from the given polynomial  `f(x)x^3 - 2x^2 + ax - b,`then rest of part of this polynomial is exactly divisible by x² − 2x − 3.

Therefore, `p(x) = x^3 - 2x^2 + ax - b - (x - 6)`

Now,

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

`x^2 - 2x - 3 = (x+1)(x-3)`

Therefore, (x + 1)(x -3)are factors of polynomial p(x).

Now,

 p(-1) = 0

And

p(3) = 0

`p(-1) = (-1)^2 + a(-1) - b-(-1-6) = 0`

                              ` = -1-2-a-b+1+6 = 0`

                              ` = -a -b+4 = 0`

                     ` a+b = 4         .......... (1)`

and

`p(3) = (3)^3 -2(3)^3 + a(3) - b(3-6) = 0`

        ` = 27 - 18 + 3a - b+3 = 0`

                                         `3a - b = -12            .......(2)`

Solving (i) and (ii) we get

 a = -2 ,b = 6