If the polynomial 6x^4 + 8x^3 -5x^2+ax + b is exactly divisible by the

1.	If the polynomial 6x^4 + 8x^3 -5x^2+ax + b is exactly divisible by the polynomial 2x^2 -5, thenfind the values of a and b.
Answer» since the the above polynomial is exactly divisble by 2x^2 -5..it means that the remainder is 0 therefore 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

If the polynomial 6x^4 + 8x^3 -5x^2+ax + b is exactly divisible by the polynomial 2x^2 -5, thenfind the values of a and b.

Answer»

since the the above polynomial is exactly divisble by 2x^2 -5..it means that the remainder is 0

therefore 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