Question

The roots of the equation \( x^4 - 20x^3 + 140x^2 - 400x + 384 = 0 \) are in

• A. Arithmetic Progression
• B. Geometric Progression
• C. Harmonic Progression
• D. Both Geometric Progression and Harmonic Progression

Hint:

For questions asking about the nature of roots (AP, GP, etc.), assuming the pattern and using the sum and product of roots from Vieta's formulas is usually the fastest method. For a quartic equation in AP, the sum of roots immediately gives you the mean value 'a'.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to determine if the roots of the given quartic polynomial follow a specific pattern like Arithmetic Progression (AP) or Geometric Progression (GP). We can use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.
Step 2: Key Formula or Approach:
Let the roots of \( Ax^4 + Bx^3 + Cx^2 + Dx + E = 0 \) be \( r_1, r_2, r_3, r_4 \). Sum of roots: \( \Sigma r_i = -B/A \). Product of roots: \( r_1 r_2 r_3 r_4 = E/A \). Let's assume the roots are in AP. They can be represented as \( a-3d, a-d, a+d, a+3d \).
Step 3: Detailed Explanation:
The given equation is \( x^4 - 20x^3 + 140x^2 - 400x + 384 = 0 \). 1. Test for Arithmetic Progression (AP):
Let the roots be \( a-3d, a-d, a+d, a+3d \). From Vieta's formulas, the sum of the roots is:
\[ (a-3d) + (a-d) + (a+d) + (a+3d) = -(-20)/1 = 20 \] \[ 4a = 20 \implies a = 5 \] So the roots are of the form \( 5-3d, 5-d, 5+d, 5+3d \).
The product of the roots is:
\[ (5-3d)(5+3d)(5-d)(5+d) = 384/1 = 384 \] \[ (25 - 9d^2)(25 - d^2) = 384 \] Let \( y = d^2 \).
\[ (25 - 9y)(25 - y) = 384 \] \[ 625 - 25y - 225y + 9y^2 = 384 \] \[ 9y^2 - 250y + 625 - 384 = 0 \] \[ 9y^2 - 250y + 241 = 0 \] We can solve this quadratic equation for \(y\). The sum of roots is \( 1+241/9 \) and product is \( 241/9 \). Let's test integer values. If \(y=1\): \[ 9(1)^2 - 250(1) + 241 = 9 - 250 + 241 = 250 - 250 = 0 \] So, \( y = 1 \) is a solution. This means \( d^2 = 1 \implies d = \pm 1 \). Let's take \( d = 1 \). The roots are:
\( a-3d = 5-3 = 2 \)
\( a-d = 5-1 = 4 \)
\( a+d = 5+1 = 6 \)
\( a+3d = 5+3 = 8 \) The roots are 2, 4, 6, 8. These are in AP. We can verify these roots with the other coefficients to be certain, and they hold true.
Step 4: Final Answer:
Since we found a consistent set of roots (2, 4, 6, 8) that are in Arithmetic Progression, this is the correct pattern.