Question

Find the value is 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

From factor theorem

If (x + 1) and (x – 1) are factors of f (x) then f (-1) = 0 and f (1) = 0 respectively.

f (-1) = 0

(-1)⁴ + a (-1)³ – 3 (-1)² + 2 (-1) + b = 0

1 – a – 3 – 2 + b = 0

b – a – 4 = 0 (i)

Similarly, 

f (1) = 0

(1)⁴ + a (1)³ – 3 (1)² + 2 (1) + b = 0

1 + a – 3 + 2 + b = 0

a + b = 0 (ii)

Adding (i) and (ii), we get

2b – 4 = 0

2b = 4

b = 2

Putting the value of b in (i), we get

2 – a – 4 = 0

a = - 2

Hence, 

a = - 2 and b = 2.