Question

Find the values of p and q so that x⁴ + px³ + 2x³ − 3x + q is divisible by (x² − 1).

Answer 1

Let  f(x) =  x⁴ + px³ + 2x³ − 3x + q and `g(x) = x^2 - 1`be the given polynomials.

We have,

`g(x)= x^2 - 1`

` = (x-1)(x+ 1)`

Here,  (x-1),(x+1)are the factor of g(x).

If f(x) is divisible by (x-1)and (x+1), then (x-1)and (x+1) are factor of f(x).

Therefore, f(1) and f(−1) both must be equal to zero.

Therefore,

`f(1) = (1)^4 + p(a)^3 + 2(1)^2 - 3(1)+q`    ......... (1)

`⇒ 1+ p + 2 - 3 + q = 0`

                              `p+q = 0`

and 

`f(-1) = (-1)^4 + p(-1)^3 + 2(- 1)^2 - 3(-1) + q = 0`

                                       ` 1-p+2 + 3 +q = 0`

                                                          `-p + q = -6         ......(2)`

Adding both the equations, we get,

`(p+q) + (-p + q) = -6`

                                `2q = -6`

                                  `q = -3`

Putting this value in (i)

`p+(-3) = 0`

                `p = 3`

Hence, the value of p and q are 3, −3 respectively.