Question

A possible value of \( b \in \mathbb{R} \) for which the equation \[ x^4 + bx^3 + 1 = 0 \] has no real root is

• A. \( -\frac{11}{5} \)
• B. \( -\frac{3}{2} \)
• C. 2
• D. \( \frac{5}{2} \)

Hint:

When solving polynomial equations with real coefficients, use discriminants or graphical methods to check for the number of real roots.

Answer 1

The Correct Option is B

Step 1: Analyze the equation.
The equation is a quartic equation with a cubic term \( bx^3 \). We can check for possible values of \( b \) that make the equation have no real roots by analyzing the discriminant or using graphical methods. Step 2: Try values of \( b \).
For \( b = -\frac{3}{2} \), the equation \( x^4 - \frac{3}{2}x^3 + 1 = 0 \) has no real roots (as verified by either the discriminant or numerical methods). Step 3: Conclusion.
The correct answer is \( \boxed{ -\frac{3}{2} } \).