HARD Polynomial question... (1 Viewer)

[its_bui]

given p(x)= x^3 + bx + 1 where b is real
find the values of b such that P(x)=0 has two complex roots.

ps: no joels or anthony s. allowed

 

[dawso]

hm.....well complex roots appear in complex cons. so it means basically "has one real root"......

think back to polynomials dawso......

 

[dawso]

wait, does the question say complex, or imaginary, cause if it says complex, all numbers are complex so this just means it can be absolutely anythin (and it will have 3...)

 

[ngai]

given p(x)= x^3 + bx + 1 where b is real
find the values of b such that P(x)=0 has two complex roots.

ps: no joels or anthony s. allowed

if b>=0, then p'(x) >= 0 so curve is increasing (or stationary), and so it cant have more than 1 real root
if b<0, then:
p'(x) = 3x²+b
p''(x) = 6x
p'(x)=0 at x=x₁ and x=x₂, where:
x₁ = sqrt(-b/3) and x₂ = -sqrt(-b/3)
also, p''(x₁)>0 and p''(x₂)<0
if u draw a picture of a cubic that isnt strictly increasing or stationary,
then to have only one real root, the local min > 0 or local max < 0
for local min > 0, this means p(x₁) > 0, which u solve to get b > -3/cubert4
for local max < 0, this means p(x₂) < 0, which gives [4b*sqrt(-b/3)]/3 > 1, which is impossible since b < 0 (and so lhs < 0)
so the answer is b>=0 or b > -3/cubert4, which is simply b > -3/cubert4
and check for silly mistakes...coz i didnt check any of that

 

[ngai]

wait, does the question say complex, or imaginary, cause if it says complex, all numbers are complex so this just means it can be absolutely anythin (and it will have 3...)

many many questions say stuff like "complex roots", or "w is a complex root of unity"
and strictly speaking, theyre all wrong since their question blows up when we take real numbers as being complex
so yes, you're correct, but the question means "non-real" and not "complex"

 

[dawso]

wen in doubt, ask ngai.... so im assuming this answer is based on there being 2 IMAGINARY roots??

 

[dawso]

ah, goodo

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[JamiL]

instead of pickin on the english of the question can we do the maths...
i let z and z-bar be the complext roots, n i get 2 solution 2 the equation, although both dont give me o as the coeffiecent of x^2... so im lost

 

[SeDaTeD]

That actually is maths, not the "english" of the problem. Definitions are very important in mathematics.

 

[JamiL]

yes the definition is very important, although we all no wot the question means, and some ppl are just be annoying