Question

if the polynomial 6x^4 + 8x^3 -5x^2 + ax +b is exactly divisible by 2x^2 -5, then find values of a and b

Answer 1

Hi
since the the above polynomial is exactly divisble by 2x^2 -5..it means that the remainder is 0
therefir we can write it as
6x^4 + 8x^3 - 5x^2 + ax + b = (2x^2 - 5)× g(x)
where g(x) is a polynomial of 2nd degree
simplifying this furthur..we can arrive at
6x^4 + 8x^3 - 5x^2 + ax + b = (√2x - √5)(√2x +√5)× g(x)
now r.h.s become 0 when x =√(5/2) and x = -√(5/2)
by substituting x =√(5/2)
6(25/4) +8(5√5/2√2) - 5(5/2) + a(√5/√2) +b =0
25 + 20(√5/√2) + a(√5/√2) +b =0 -----(1)
by substituting x = -√(5/2)
6(25/4) - 8(5√5/2√2) - 5(5/2) - a(√5/√2) +b =0
25 - 20(√5/√2) - a(√5/√2) +b =0 -----(2)
(1) + (2)
50 +2b =0b = - 25

(1) - (2)
40(√5/√2) + 2a(√5/√2) =0a = -20

hope this will help you :)