If α,β,γ be the roots of the equation ax3+bx2+cx+d=0 , then the equation whose roots are −α,−β,−γ is
Let's find the product of the roots of the equation ax^3 + bx^2 + cx + d = 0 using Vieta's formulas:
αβγ = -d/a
Now, let's find the equation whose roots are -α, -β, and -γ. By applying Vieta's formulas again, we have:
Sum of the roots: (-α) + (-β) + (-γ) = -(α + β + γ) = -(-b/a) = b/a
Product of the roots: (-α)(-β)(-γ) = -αβγ = -(-d/a) = d/a
Therefore, the equation whose roots are -α, -β, and -γ is:
x^3 - (b/a)x^2 + 0x + (d/a) = 0
Simplifying,
ax^3 - bx^2 + cx - d = 0
So, the equation whose roots are -α, -β, and -γ is ax^3 - bx^2 + cx - d = 0.