Question

Find the values of a and b so that (x + 1) and (x − 1) are factors of x⁴ + ax³ − 3x² + 2x + b.

Answer 1

Let  f(x) = x⁴ + ax³ − 3x² + 2x + b be the given polynomial.

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

Therefore,

`f(-1) = (-1)^4 + a(-1)^3 -3(-1)^2 +2(-1) + b = 0    ..... (1)`

                                              ` 1-a - 3- 2 + b = 0`

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

                                          and

`f(1) = (1)^4 + a(1)^3 +2(1) +b = 0`

                              `1 + a-3 + 2 = 0`

                                              `a+b = 0            .....(2)`

Adding equation (i) and (ii), we get

                          `2b = 4`

                            `b = 2`

Putting this value in equation (i), we get,

-a + 2 = 4

a = -2

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