Polynomials Question... (1 Viewer)

[Michaelmoo]

let x = a be a root of the polynomial p(x) = x^4 + Ax^3 + Bx^2 + Ax + 1

where (2+B)^2 is not equal to 4A^2

(i) show that a cannot be o, -1 or 1

(ii) show that x = 1/a is a root

(iii) Deduce that if a is a multiple root, then its multiplicity is 2 and 4B = 8 + A^2

Ok i, and ii are straight forward. For iii, Anyone know of a wat way to approach this?

Also, you can begin by assuming a is of AT LEAST multiplicity 2 right?

Thanks in advance.

 

[addikaye03]

For part ii, can u just prove the recipricol root theorem? and than state that since x=a, x=1/a must also be a root? wiki it if u dont know what im talking about, other i will type it up

 

[tommykins]

let x = a be a root of the polynomial p(x) = x^4 + Ax^3 + Bx^2 + Ax + 1

where (2+B)^2 is not equal to 4A^2

(i) show that a cannot be o, -1 or 1

(ii) show that x = 1/a is a root

(iii) Deduce that if a is a multiple root, then its multiplicity is 2 and 4B = 8 + A^2

Ok i, and ii are straight forward. For iii, I've done it but my method takes forever. ANyone have a quick way of approaching this?

Also, you can begin by assuming a is of AT LEAST multiplicity 2 right?

Thanks in advance.

Differentiate p(x) and sub in x = a.

solve p(x) and p'(x) simultaneously.

i havent tried it myself but the multiple root should give u the idea to differentiate

 

[Michaelmoo]

Differentiate p(x) and sub in x = a.

solve p(x) and p'(x) simultaneously.

i havent tried it myself but the multiple root should give u the idea to differentiate

Tried that. It's not that simple though. Doesn't work out.

 

[hotfiree]

When the hell will you use this in life?

 

[Michaelmoo]

When the hell will you use this in life?

They never taught you real world applications of calculus at school?

 

[lolokay]

When the hell will you use this in life?

when will you use problem solving/reasoning skills in life?

 

[lolokay]

for the question, just use sum/product of roots

 

[Trebla]

Well if a is a multiple root, then it should occur at least twice. Since a =/= 1/a because a =/= 1, -1, the roots a and 1/a are distinct. Let the forth unknown root be β.
Product of roots: a²(1/a)β = 1
=> β = 1/a
This means there are two double roots a and 1/a
So yeah, that was a weird way to show a had mulitplicity 2...lol

 

[Michaelmoo]

when will you use problem solving/reasoning skills in life?

What exactl do u mean by "use"

 

[lolokay]

What exactl do u mean by "use"

err I meant the same thing the other guy meant

i was just pointing out to him that even though you're most likely never going to solve a problem like this in life, you will most likely be utilising the problem solving skills gained in maths

 

[Michaelmoo]

err I meant the same thing the other guy meant

i was just pointing out to him that even though you're most likely never going to solve a problem like this in life, you will most likely be utilising the problem solving skills gained in maths

Ahhh ok.

 

[gurmies]

Symmetric coefficients are nice...

 

[Michaelmoo]

Symmetric coefficients are nice...

Yer I tried working with it like that, couldnt find anyhing. Any suggestions?

 

[lolokay]

^ if you still haven't got the solution, like i said just use sum/product of roots

you can find the other root + show multiplicity is 2 using what trebla said

then you get an expression for A and B using sum of roots 1 and 2 at a time, then just show that 4B = A² + 8