If x^3+3px+q has a factor of the form x^2-2ax+a^2 show that q^2+4p^3=0

Answer 1

a^2 - 2ax + a^2 = (x - a)^2

so x^3 + 3px + q = (x-a)(x-a)(x-k)
the last term of (x-a)(x-a)(x-k) is -ka^2, so
-ka^2 = q ----> k = -q/a^2
and x^3 + 3px + q = (x-a)(x-a)(x+q/a^2)
= (x+q/a^2)(x^2 - 2ax + a^2)
= x^3 - 2ax^2 + a^2 x + (q/a^2)x^2 - (2q/a)x + q
= x^3 + (-2a + q/a^2)x^2 + (a^2 - 2q/a)x + q

now match them up:
x^3 = x^3, ok
q = q, ok
no x^2 term in x^3 + 3px + q , so q/a^2 - 2a = 0 ----> a^3 = q/2
a^2 - 2q/a = 3p
a^2 = 3p + 2q/a
a^3 = 3ap + 2q
argghhhh, must have messed up to end up with "a" on the right side,

Should have written it out on paper first.
See if you can find it, I trust my method

Answer 2

Sir/Madam, I think that , in first step X²-2ax+a²=(x-a) ²