Question

The smallest positive root of the equation \[ x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0 \] lies in the range

• A. \( 0<x \leq 2 \)
• B. \( 2<x \leq 4 \)
• C. \( 6 \leq x \leq 8 \)
• D. \( 10 \leq x \leq 100 \)

Hint:

For polynomial equations, substitution of values within the given intervals can help identify the correct range of roots efficiently.

Answer 1

The Correct Option is A

Step 1: The given equation is \[ x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0 \] To find the smallest positive root, we can use trial values within the given options. Step 2: Substituting values in the equation: \[ \text{For } x = 1: \quad (1)^5 - 5(1)^4 - 10(1)^3 + 50(1)^2 + 9(1) - 45 = 0 \] Since the equation satisfies \( x = 1 \), the smallest root lies in the range \( 0<x \leq 2 \). Step 3: Checking for higher values such as \( x = 3, 7, 12 \), they do not satisfy the equation. Hence, the smallest root is within the interval \( (0,2] \). Conclusion: The correct answer is option (A) \( 0<x \leq 2 \).