Question

Find the values of a and b, if x² − 4 is a factor of ax⁴ + 2x³ − 3x² + bx − 4.

Answer 1

Let `f(x) = ax^4 + 2x^3 - 3x^2 + bx - 4` and `g(x) = (x^2 - 4)`be the given polynomial.

We have,

`g(x) = (x^2 - 4)`

        ` = (x-2) ( x + 2)`

`{because a^2 - b^2 = (a-b )(a+b)}`

 ⇒ (x − 2), (x + 2) are the factors of g(x).

By factor theorem, if (x − 2) and (x + 2) both are the factor of f(x)

Then f(2) and f(−2) are equal to zero.

Therefore,

`f(2) = a(2)^4 + 2(2)^3 - 3(2)^2 + b (2) - 4 =0`

`16a + 16 - 12 + 2b - 4 =0`

`16a+ 2b = 0`

` 8a + b =0`                 ...(i)

and

`f(2) = a(-2)^4 + 2(- 2)^3 - 3(-2)^2 + b (-2) - 4 =0`

`16a - 16 - 12 - 2b - 4 =0`

                `16a- 2b = 32 = 0`

                            ` 8a - b =16`              ...(ii)     

Adding these two equations, we get

`(8a + b) + (8a - b) = 16`

                             ` 16a = 16`

                                   `a= 1`

Putting the value of a in equation (i), we get

`8 xx 1+ b = 0`

               ` b = -8`

Hence, the value of a and b are 1, − 8 respectively.