The equation whose roots are those of equation ` x^4 - 3x^3 +5x^2 -2=0` with contrary sign is

Options

A. `x^4 +3x^3 +5x^2 -2=0`
B. `x^4 +3x^3 +x^2 +7x -2=0`
C. `x^4 -3x^3 +8x ^2 +4=0`
D. `10x^4 -13x^2 +40=0`

Answer

To find the equation whose roots are the contrary (opposite) of the roots of the given equation \( x^4 - 3x^3 + 5x^2 - 2 = 0 \), we can follow these steps:

### Step 1: Identify the given equation
The given equation is:
\[ x^4 - 3x^3 + 5x^2 - 2 = 0 \]

### Step 2: Understand the meaning of "contrary sign"
The phrase "contrary sign" means that if \( r \) is a root of the original equation, then \( -r \) will be a root of the new equation. This will affect the odd-degree terms (the terms with \( x^3 \) and \( x^1 \)) in the polynomial.

### Step 3: Change the signs of the odd-degree terms
The original polynomial has the following terms:
- \( x^4 \) (even degree, remains the same)
- \( -3x^3 \) (odd degree, change the sign to \( +3x^3 \))
- \( 5x^2 \) (even degree, remains the same)
- \( -2 \) (constant term, remains the same)

Thus, the new polynomial will be:
\[ x^4 + 3x^3 + 5x^2 - 2 = 0 \]

### Step 4: Write the final equation
The equation whose roots are the contrary of the roots of the given equation is:
\[ x^4 + 3x^3 + 5x^2 - 2 = 0 \]

### Final Answer
The required equation is:
\[ x^4 + 3x^3 + 5x^2 - 2 = 0 \]

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