Question

If the equation \(x^4 + 7x^3 + 18x^2 + 20x + 8 = 0\) has a repeated root, then that repeated root is:

• A. \( \mathbf{-2} \)
• B. \( -1 \)
• C. \( -3 \)
• D. \( -4 \)

Hint:

To find repeated roots, check which root is common between the polynomial and its derivative.

Answer 1

The Correct Option is A

Let \( f(x) = x^4 + 7x^3 + 18x^2 + 20x + 8 \). Since the polynomial has a repeated root, it must be a common root of \( f(x) \) and \( f'(x) \). Compute derivative: \[ f'(x) = 4x^3 + 21x^2 + 36x + 20 \] Use Euclidean algorithm or factorization approach. Try rational root theorem: test \( x = -2 \): \[ f(-2) = 16 - 56 + 72 - 40 + 8 = 0 \Rightarrow \text{So } x = -2 \text{ is a root} \] Now check if it’s repeated: \[ f'(-2) = -32 + 84 - 72 + 20 = 0 \Rightarrow \text{So } x = -2 \text{ is also a root of } f'(x) \] Hence, \( x = -2 \) is a repeated root.