Question

If \(α,β,γ\) are the roots of \(4x^3-6x^2+7x+3=0\), then the value of \(αβ+βγ+γα\) is

• A. \(-\frac 74\)
• B. \(\frac 74\)
• C. \(-\frac 23\)
• D. \(-\frac 32\)

Answer 1

The Correct Option is B

Given the cubic polynomial 4x³ - 6x² + 7x + 3 = 0 with roots α, β, γ.

For any cubic polynomial of the form ax³ + bx² + cx + d = 0, the following relationships hold for its roots:

• Sum of roots (α + β + γ) = -b/a
• Sum of product of roots taken two at a time (αβ + βγ + γα) = c/a
• Product of roots (αβγ) = -d/a

For the given polynomial 4x³ - 6x² + 7x + 3 = 0:

• a = 4 (coefficient of x³)
• b = -6 (coefficient of x²)
• c = 7 (coefficient of x)
• d = 3 (constant term)

We need to find the value of αβ + βγ + γα, which is given by c/a.

Calculating this:

αβ + βγ + γα = c/a = 7/4

Therefore, the value of αβ + βγ + γα is 7/4.

The correct answer is: (2) 7/4