If the roots of ` x^4 +5x^3 - 30 x^2 - 40 x+64=0` are in G.P then roots of ` x^4 - 5 x^3 - 30 x^2 + 40 x+ 64 =0` are in

Options

A. A.P
B. G.P
C. H.P
D. A.G.P

Answer

To solve the problem, we need to analyze the given polynomial equations and their roots.

### Step 1: Understand the first equation
We start with the polynomial equation:
\[ x^4 + 5x^3 - 30x^2 - 40x + 64 = 0 \]
We are informed that the roots of this equation are in Geometric Progression (G.P).

### Step 2: Analyze the second equation
Next, we consider the second polynomial:
\[ x^4 - 5x^3 - 30x^2 + 40x + 64 = 0 \]

### Step 3: Substitute \( x \) with \( -x \) in the first equation
To relate the two equations, we substitute \( x \) with \( -x \) in the first equation:
\[ -x^4 + 5(-x^3) - 30(-x^2) - 40(-x) + 64 = 0 \]
This simplifies to:
\[ -x^4 - 5x^3 - 30x^2 + 40x + 64 = 0 \]
Multiplying through by -1 gives:
\[ x^4 + 5x^3 + 30x^2 - 40x - 64 = 0 \]

### Step 4: Compare the modified equation with the second equation
Notice that the modified equation is:
\[ x^4 - 5x^3 - 30x^2 + 40x + 64 = 0 \]
This is exactly the second equation we started with.

### Step 5: Conclude the relationship of roots
Since the roots of the first equation are in G.P., and the second equation is obtained by substituting \( x \) with \( -x \), the roots of the second equation will also maintain a specific relationship.

### Step 6: Identify the type of progression
When the roots of a polynomial are transformed by substituting \( x \) with \( -x \), the nature of the roots changes. If the original roots are in G.P., the new roots will be in Arithmetic Progression (A.P).

### Final Answer
Thus, the roots of the equation \( x^4 - 5x^3 - 30x^2 + 40x + 64 = 0 \) are in **Arithmetic Progression (A.P)**.

---