Polynomial Questions (1 Viewer)

[Kutay]

Hey all, i have been tackling some polynomial questions and have come across i just cant seem to get the answers to, i was wondering if anyone could help me out.
Thank you

"Given two roots of 2x^3 - kx^2 + 8 = 0 are equal, find k."

"Form the cubic equation whos roots are 2, 3 and 5"

" If x^3 - 8x^2 + kx -12 = 0 has one root equal to the sum of the other two, find the value of k"

In regards to this next one, i know how to do the Arithmetic Progression one but not sure with the GP one,

"Solve the equation x^3 - 14x^2 + 56x - 64 = 0 if the roots are in geometric progression."

 

[香港!]

Hey all, i have been tackling some polynomial questions and have come across i just cant seem to get the answers to, i was wondering if anyone could help me out.
Thank you

"Given two roots of 2x^3 - kx^2 + 8 = 0 are equal, find k."

"Form the cubic equation whos roots are 2, 3 and 5"

" If x^3 - 8x^2 + kx -12 = 0 has one root equal to the sum of the other two, find the value of k"

In regards to this next one, i know how to do the Arithmetic Progression one but not sure with the GP one,

"Solve the equation x^3 - 14x^2 + 56x - 64 = 0 if the roots are in geometric progression."

Hello Kutay!!

"Given two roots of 2x^3 - kx^2 + 8 = 0 are equal, find k."
let the roots be A, A and B since two of the roots are equal
A+A+B=2A+B=k\2
A(A)+A(B)+A(B)=A²+2AB=0
.: A=-2B
AAB=-8\2=-4
(-2B)²B=-4
4B³=-4
B³=-1
.: B=-1 and hence A=2
2(2)+(-1)=k\2
k=6

"Form the cubic equation whos roots are 2, 3 and 5"
(x-2)(x-3)(x-5)=0
expand then u have it

" If x^3 - 8x^2 + kx -12 = 0 has one root equal to the sum of the other two, find the value of k"

Let roots be A, B, C and choose B+C=A
so A+B+C=8
but B+C=A, .: 2A=8
A=4
so (4)^3-8(4)^2+k(4)-12=0
.: k=19

"Solve the equation x^3 - 14x^2 + 56x - 64 = 0 if the roots are in geometric progression."
u can let roots be A\B, B, AB den do it
but im gonna cheat
By inspection x=2, x=4, x=8 LoLz

 

[Riviet]

By inspection x=2, x=4, x=8 LoLz

haha, nice one 香港!
Anyway how are A\B, B, AB in a GP? I think you meant A/B, A, AB ^_^

 

[香港!]

^yeap hehehe