Question

If both x + 1 and x − 1 are factors of ax³ + x² − 2x + b, find the values of a and b.

Answer 1

Let  f(x) = ax³ + x² − 2x + b be the given polynomial.

By factor theorem, if (x+1) and  (x-1)both are factors of the polynomial f (x). if f(−1) and f(1) both are equal to zero.

Therefore,

`f(-1) = a(-1)^3 + (-1)^2 - 2 (-1) +b= 0`

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

                                                        `-a+b = -3   ....(1)`

And

`f(1) = a(1)^3 + 1(1)^3 - 2(1) +b = 0`

`a+1 -2 + b = 0`

               `a+b = 1     ......... (2)`

Adding (i) and (ii), we get

 2b =-2

b =-1

And putting this value in equation (ii), we get,

a = 2

Hence, the value of a and b are 2 and −1 respectively.