Question

without actual division, prove that x^4+2x^3+2x-3 is exact;y divisible by x^2+2x+3.



pls, give an elaborative answer I'm not able to do it.

Answer 1

Let f(x) = x4 + 2x3 -2x2 + 2x - 3

g(x) = x2 + 2x - 3

= x(x + 3) - 1(x + 3) = (x - 1) (x + 3)

Now f(x) will be exactly divisible by g(x) if it is exactly divisible by (x - 1) as well as (x + 3)

i.e. if f(1) = 0 and f ( -3) = 0

Now f(1) = 14 + 2.13 -2.12 + 2.1 - 3

= 1 + 2 - 2 + 2 - 3 = 0

=> (x - 1) is a factor of f(x)

f ( -3) = (-3)4 + 2.(-3)3 -2.(-3)2 + 2.(-3) - 3

= 81 - 54 - 18 - 6 - 3 = 0

=> (x + 3) is a factor of f(x).

=> (x - 1) (x + 3) divides f (x) exactly

Therefore, x2 + 2x - 3 is a factor of f(x)