Find a necessary and sufficient condition on a, b, c that the roots of x³ + ax² + bx + c = 0 are in A.P.?

Let the roots be m-d, m and m+d

=> m-d + m + m+d = - a

=> m = - a/3

m(m-d) + m(m+d) + (m-d)(m+d) = b

=> 3m^2 - d^2 = b

=> d^2 = a^2/3 - b

=> roots are

-a/3 - √(a^2/3-b), - a/3 and -a/3 + √(a^2/3-b)

=> product of the roots

(-a/3 - √(a^2/3 - b) * (-a/3) * (-a/3 + √(a^2/3 - b) = - c

=> - (a/3) * (a^2/9 - a^2/3 + b) = - c

=> (a/3) (b - 2a^2/9) = c

=> c = (a/27) (9b - 2a^2)

This is the necessary and sufficient condition for the roots to be in A.P.

Sufficient because the roots of the equation with the above value of c are the ones found as above for which refer to the following Wolfram Alpha link:

http://www.wolframalpha.com/input/?i=x%C2%B3+%2B+a...

x^5-7x^4+Ax^3+Bx^2+Cx+D=0? If a million + 2i and a million +3i are its roots, then a million - 2i and a million - 3i are additionally its roots because of the fact complicated roots ensue in pairs of complicated conjugate numbers. enable the 5th root be a, then the sum of the roots = 4 + a =7 which consequences up in a = 3. this might desire to fulfill the given equation , so : 27 A + 9 B + 3 C +D = 324..............(a million) made from the roots =(a million+2i) (a million-2i)(a million+3i)(a million-3i) 3 = -D or 5*10* 3 = - D or D = - a hundred and fifty answer Sum of the products of roots taken 2 at a time = 3(a million+2i) + 3( a million-2i) +3(a million+3i)+ 3 (a million-3i) +(a million+2i)(a million-2i) +(a million+3i)(a million-3i) +(a million+2i)(a million+3i) +(a million+2i) (a million-3i)+ (a million-2i)(a million+3i) + (a million-2i)(a million-3i) = A So : A = 3*4 +5 +10 +2(a million+2i) +2(a million-2i) = 31. Or A =31Answer in addition you will discover B and C, through taking sum of goods of roots taken 3 at a time and four at a time..

Let the roots be m-d, m and m+d

then as per vieta's formula http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formul...

1)

sum of roots = m-d + m + m+d = - a or 3m = - a

2)

b= m(m-d) + m(m+d) + (m-d)(m+d) = 3m^3 - d^2 or d^2 = 3m^2 - b or (a^2)/3- b

3)

product of roots = m(m-d)(m+d) = m(m^2 - d^2) = - c

or (-a/3)((a/3)^2 - (a^2/3 -b) = -c

or(- a/3)(2a^2/9 - b) = - c

or (a/3) (b - 2a^2/9) = c

or (a/27) (9b - 2a^2) = c

above condition is necessary and sufficient

refer to http://math-kali.blogspot.com/2011/12/find-suffici...