Question

If x^2-1 is a factor of f(x)=x^4+ax+b find a and b

Answer 1

on dividing x⁴+ ax + b by (x²-1) we get the remainder of ax +(b+1)
ax+b+1 should be equal to zero as x²- 1 is its factor
therefore ax+b+1 can only be zero when a= 0 and b = -1

Answer 2

if x²-1 is a factor then (x+1) and (x-1) will also be the factor of the polynomial,

then

f(-1)=0,

(-1)⁴+a(-1)+b=0,

1-a+b=0,

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

also

f(1)=0,

(1)⁴+a(1)+b=0,

1+a+b=0,

a+b=-1 --------------eq(2),

now solve these two equations, We get

a=0,

b=-1