Further Maths problem, Help!!

The roots of a cubic equation are α, β and γ. If Σα=3, Σαβ=7/2 and αβγ=-5 state the cubic equation. Is it possible? And if yes how do you do it, because I have no clue?

The roots of a cubic equation are α, β and γ. If Σα=3, Σαβ=7/2 and αβγ=-5 state the cubic equation. Is it possible? And if yes how do you do it, because I have no clue?

This is concerning the roots of polynomials. Some online research/further reading as suggested with the link above is a good idea.

lets say the equation is of the form
ax^3 + bx^2 + Cx + d =0

The sums of the root relate to the coefficients
Σ⍺ is -b/a ------ [⍺+β+ɣ]
Σ⍺β is c/a ----- [⍺β+βɣ+⍺ɣ]
Σ⍺βɣ is -d/a ----- [note this is obviously the same as ⍺βɣ]

Using this you form three simultaneous equations:
1) -b/a=3
2) c/a=7/2
3) -d/a=-5

At this stage you have 3 equations with 4 variables and just need to eliminate.
For these equations easiest thing to do is to eliminate using a mathematical statement.

that gives you
a=-b/3=2c/7=-d/-5

Cleaning this up
Statement is 35b=-30c=-21d

Now you must find the lowest common multiple of 35, 30 and 21 (negative or positive it doesn't matter) Which is 210 by prime factorization, or just trial and error.


Figure out what b c and d are using the statement. 35b=-30c=-21d=210
b=6
c=-7
d=-10

substitute back into equation 1) to find a

-6/a=3
so a=-2

Check this works with the other equations 2) and 3) which it does !! thank god
and then you know the cubic.

-2x^3 + 6x^2 -7x -10
OR if you used -210 with the mathematical statement
2x^3 -6x^2 +7x +10

These two cubics have the same roots so both answers are correct you can state either one. They are not the same equation obviously lol. There may be other ways however this worked for me. Hope it helps you out.