Polynomial/roots question [Mathematics Preliminary Extension 1] (1 Viewer)

[Nktnet]

The equation x³-bx²+cx-d = 0, where d>0, has roots α, ß and αß. Show that (c+d)² = (b+1)²d.


New to forums and not sure if this is how things are done. Sorry in advance!
And thanks!

 

[HoldingOn]

The equation x³-bx²+cx-d = 0, where d>0, has roots α, ß and αß. Show that (c+d)² = (b+1)²d.


New to forums and not sure if this is how things are done. Sorry in advance!
And thanks!

How you have done it is just fine- just post any questions and in time I'm sure they will be answered.

For this particular problem:

Derive equations for the sum, sum for two at a time, and product of roots. [I will use a, B and aB as the roots] These will give you:
1. (aB)^2=d
2. a+B+aB= b
3. aB(1+a+B)=c

Adding 1 and 3 together we get

c+d= aB+a^2 B+ aB^2 + (aB)^2

Factorising:

c+d= aB(1+a+B+aB)

Squaring both sides we get

(c+d)^2= (aB)^2 (1+a+B+aB)^2

Now we sub in our initial pro-numerals

(c+d)^2 = d(1+b)^2

 

[Nktnet]

Thank you so much!
That makes a lot of sense and was really easy to follow.