Question

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

Answer 1

Let, f (x) = ax⁴ + 2x³ - 3x² + bx - 4 and g (x) = x² – 4

We have,

g (x) = x² – 4

= (x – 2) (x + 2)

Given,

g (x) is a factor of f (x)

(x – 2) and (x + 2) are factors of f (x)

From factor theorem if (x – 2) and (x + 2) are factors of f (x) then f (2) = 0 and f (-2) = 0 respectively.

f (2) = 0

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

16a – 16 – 12 + 2b – 4 = 0

16a + 2b = 0

2 (8a + b) = 0

8a + b = 0 (i)

Similarly,

f (-2) = 0

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

16a – 16 – 12 - 2b – 4 = 0

16a - 2b – 32 = 0

16a – 2b – 32 = 0

2 (8a - b) = 32

8a – b = 16 (ii)

Adding (i) and (ii), we get

8a + b + 8a – b = 16

16a = 16

a = 1

Put a = 1 in (i), we get

8 x 1 + b = 0

b = -8

Hence, 

a = 1 and b = -8.