Standard teaching says roots are (x-A)(x-B)(x-C)=0, so the roots of ax³+bx²+cx+d=0 are such that A+B+C=-b/a, AB+BC+CA=c/a and ABC=-d/a. We can play algebraic games with these to produce related equations. What is not clear is why we would want to do this, beyond proving we can do the manipulation.
Textbooks prefer to use alpha, beta and gamma. I have issues with reproducing those characters. Also, I noted that users of Mandarin had issues with writing alpha and gamma in readable ways.
α, β, γ: I am impressed with neither the alpha nor gamma. Unicode U+0381 to 3
𝛂, 𝛃, 𝛄 are better, Unicode U+1D62 to 4
Show that (A+B+C)² = A²+ B²+ C² +2(AB+BC+CA)
Show that (A+B+C)³ = A³+ B³+ C³ +3(A²B+B²C+C²A+ AB²+BC²+CA²) + 6ABC
Show that (AB+BC+CA)² = (AB)²+ (BC)²+ (CA)² +2(ABC)(A+B+C)
Exercise:
1. Find the related equation with roots 1/A, 1/B, 1/C.
2. Find the related equation with roots kA, kB, kC.
If it helps, check using k=2.
3. Find the related equation with roots 1/AB, 1/BC, 1/CA.
4. Find the related equation with roots A², B², C².
5. Find the related equation with roots AB, BC, CA.
6. Find the related equation with roots A+B, B+C, C+A (hard).
I thought it might be intgersting to discover where this techniques has use, so I googled ‘related equations’. A first page hit was this page, which tells me that the topic has other names. Properly, the A-level syllabus, quoting OCR here: 4.05a, Roots of equations, a) Understand and be able to use the relationships ... to obtain an equation whose roots are related to those of the original equation...
Look at https://plus.maths.org/content/101-uses-quadratic-equation. Do read more than the first page.
See also the page Related Equations, adjacent.
1. dx³+cx²+bx+a=0
2. ax³+kbx²+k²cx+k³=0
3. d²x³-b²x²+acx-a²=0
4. a²x³-(b²-2ac)x²+(c²-2bd)x-d²=0
5. a²x³-acx²+bdx-d²=0
6. Σr = -2c/a, Σrs = (b²+ac)/a2 Σ rst = -bc/a² –ad/a². So a²x³+2acx²+(b²+ac)/x+(bc+ad)=0. I think.
