Step 1: Concept Identification: The objective is to ascertain if the roots of the provided quartic polynomial exhibit an Arithmetic Progression (AP) or Geometric Progression (GP) pattern. Vieta's formulas, which establish relationships between polynomial coefficients and their root sums/products, are to be employed.
Step 2: Methodological Framework: For the quartic equation \( Ax^4 + Bx^3 + Cx^2 + Dx + E = 0 \), let the roots be \( r_1, r_2, r_3, r_4 \).
Sum of roots: \( \Sigma r_i = -B/A \).
Product of roots: \( r_1 r_2 r_3 r_4 = E/A \).
Hypothesize roots in AP as \( a-3d, a-d, a+d, a+3d \).
Step 3: Detailed Analysis: The given polynomial is \( x^4 - 20x^3 + 140x^2 - 400x + 384 = 0 \).
1. Arithmetic Progression (AP) Verification:
Assume roots are \( a-3d, a-d, a+d, a+3d \).
Applying Vieta's formulas for the sum of roots:
\[ (a-3d) + (a-d) + (a+d) + (a+3d) = -(-20)/1 = 20 \]
\[ 4a = 20 \implies a = 5 \]
Thus, the roots are structured as \( 5-3d, 5-d, 5+d, 5+3d \).
The product of the roots is:
\[ (5-3d)(5+3d)(5-d)(5+d) = 384/1 = 384 \]
\[ (25 - 9d^2)(25 - d^2) = 384 \]
Substitute \( y = d^2 \):
\[ (25 - 9y)(25 - y) = 384 \]
\[ 625 - 25y - 225y + 9y^2 = 384 \]
\[ 9y^2 - 250y + 625 - 384 = 0 \]
\[ 9y^2 - 250y + 241 = 0 \]
Solving this quadratic equation for \( y \). Using the quadratic formula or by inspection, we find that \( y = 1 \) is a solution:
\[ 9(1)^2 - 250(1) + 241 = 9 - 250 + 241 = 0 \]
Therefore, \( y = 1 \), implying \( d^2 = 1 \), so \( d = \pm 1 \).
Selecting \( d = 1 \), the roots are:
\( a-3d = 5-3(1) = 2 \)
\( a-d = 5-1 = 4 \)
\( a+d = 5+1 = 6 \)
\( a+3d = 5+3(1) = 8 \)
The roots are 2, 4, 6, 8. This sequence forms an Arithmetic Progression. Verification with other polynomial coefficients confirms the validity of these roots.
Step 4: Conclusion: A consistent set of roots (2, 4, 6, 8) forming an Arithmetic Progression has been identified, confirming this as the correct pattern.