Question

If (x-2) and (x+2) are factors of ax⁴+2x-3x²+bx-4.Find the value of a+b.
Plz give the ans and explanation

Answer 1

if (x + 2) is a factor of ax⁴ + 2x - 3x² + bx - 4 
then x + 2 = 0 and x = - 2 
ax⁴ + 2x - 3x² + bx - 4 = 0 
substituting  x = - 2 gives
16a - 4 - 12 + bx - 4 = 0 
after simplifying
16a - 2b = 20 
or 
8a - b = 10 
again,
if (x - 2) is a factor of ax⁴ + 2x - 3x² + bx - 4 
then x - 2 = 0 and x = 2
ax⁴ + 2x - 3x² + bx - 4 = 0 
substituting  x = 2 gives
16a + 4 - 12 + bx - 4 = 0 
which simplifies as
16a - 2b = 12 
or 
8a + b = 6 
⇒ the equations are 
8a - b = 10 
8a + b = 6 
from here we get
a=1
b= -2
a+b= 1+(-2) = -1
∴ for ax⁴ + 2x³ - 3x² + bx - 4  it'll be
by substituting x = - 2 
16a - 16 - 12 - 2b - 4 = 0 
and simplifying
8a - b = 16 
again,
by substituting x = 2 
16a + 16 - 12 + 2b - 4 = 0 
after simplyfying
8a + b = 0 
∴ we get 
a = 1 
b = - 8 
a + b = - 7