How do I prove that 2b^{2}-3abc+da^{2}=0 if the roots of the polynomial f(x)=ax^{3}+3bx^{2}+3cx+d are

Adrienne Rowe

Adrienne Rowe

Answered question

2022-03-02

How do I prove that
2b23abc+da2=0
if the roots of the polynomial
f(x)=ax3+3bx2+3cx+d
are in arithmetic progression?

Answer & Explanation

disgynna0vv

disgynna0vv

Beginner2022-03-03Added 3 answers

Step 1
Let xy, x, x+y be roots of the given equation.
1) xy+x+x+y=3bax=ba
2) (xy)x+x(x+y)+(x+y)(xy)=3ca
3x2y2=3ca
y2=3(b2ac)a2
3) (xy)x(x+y)=da
x(x2y2)=da
ba×b2{a2}3(b2ac)a2=da
2b33abc+da2=0

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