Question

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

Answer 1

Answer:

(a,b) = (1,-8)

Step-by-step explanation:

$x^{2} - 4 = (x+2)(x-2)$

now the polynomial has two known roots i.e., x= -2 or 2

substitute x = 2

$ax^{4} + 2x^{3} - 3x^{2} + bx - 4\\a.2^(4) + 2 .2^{3} - 3.2^{2} + 2b - 4\\16a + 16 - 12 + 2b - 4 = 0\\\\16a+2b = 0$ ---- (i)

substitute x = -2

$ax^{4} + 2x^{3} - 3x^{2} + bx - 4\\a.(-2)^(4) + 2 .(-2)^{3} - 3.(-2)^{2} - 2b - 4\\16a - 16 - 12 - 2b - 4 = 0\\\\16a- 2b = 32$---(ii)

solve (i) and (ii),

32a = 32

a = 1

substitute a = 1 in (i)

16 + 2b = 0;

2b = -16

b = -8