Question

If the roots of the cubic equation x^3 + bx^2 + cx + d are r₁, r₂, and r₃, then the value of b is:

a) -(r₁ + r₂ + r₃)
b) (r₁ + r₂ + r₃)
c) -(r₁ * r₂ * r₃)
d) (r₁ * r₂ * r₃)

Answer 1

The value of 'b' in the cubic equation x^3 + bx^2 + cx + d, given roots r₁, r₂, and r₃, is the negative sum of the roots, which is option a) -(r₁ + r₂ + r₃).

Step-by-step explanation:

If the roots of the cubic equation x^3 + bx^2 + cx + d are r₁, r₂, and r₃, then according to Viète's formulas, the sum of the roots taken one at a time is equal to -b/a. In a cubic equation where the coefficient a of x^3 is 1 (as it is in the given equation), the value of b is simply the negative sum of the roots. Therefore, the correct answer is a) -(r₁ + r₂ + r₃).

For example, if the roots of the equation are 2, 3, and 4, then the value of b would be -9, since -(2+3+4) equals -9