Question

if the polynomial 6 x power 4 + 8 x power 2 minus 5 square + a x + b is exactly divisible by the polynomial 2 x power 2 - 5 then find the value of a and b

Answer 1

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 =0
b = - 25

(1) - (2)

40(√5/√2) + 2a(√5/√2) =0
a = -20

Hope it helps you
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