Question

The cubic equation whose roots are the squares of the roots of the equation \( x^3 - 2x^2 + 3x - 4 = 0 \) is

• A. \( x^3 + 2x^2 + 7x - 16 = 0 \)
• B. \( x^3 + 2x^2 - 7x - 16 = 0 \)
• C. \( x^3 - 2x^2 - 7x + 16 = 0 \)
• D. \( x^3 - 2x^2 + 7x + 16 = 0 \)

Hint:

When finding a new polynomial whose roots are transformations of the roots of a given polynomial, use Vieta's formulas and algebraic identities to compute the new coefficients efficiently.

Answer 1

The Correct Option is B

Step 1: Find the roots of the given cubic equation. 
The given equation is \( x^3 - 2x^2 + 3x - 4 = 0 \). Let the roots be \( \alpha, \beta, \gamma \). Using Vieta's formulas for a cubic equation \( x^3 + ax^2 + bx + c = 0 \): 
\( \alpha + \beta + \gamma = -a = -(-2) = 2 \) 
\( \alpha\beta + \beta\gamma + \gamma\alpha = b = 3 \)
\( \alpha\beta\gamma = -c = -(-4) = 4 \)
We need to find the cubic equation whose roots are \( \alpha^2, \beta^2, \gamma^2 \). 
Step 2: Form the new cubic equation. 
The new cubic equation with roots \( \alpha^2, \beta^2, \gamma^2 \) can be written as: 
\[ (x - \alpha^2)(x - \beta^2)(x - \gamma^2) = 0 \] This expands to: 
\[ x^3 - (\alpha^2 + \beta^2 + \gamma^2)x^2 + (\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2)x - \alpha^2\beta^2\gamma^2 = 0 \] We need to compute the coefficients using the relationships from Vieta's formulas. 
Step 3: Compute \( \alpha^2 + \beta^2 + \gamma^2 \). 
Using the identity: 
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \] Substitute the known values: 
\[ \alpha^2 + \beta^2 + \gamma^2 = (2)^2 - 2(3) = 4 - 6 = -2 \] 
Step 4: Compute \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 \). 
First, find \( (\alpha\beta)^2 + (\beta\gamma)^2 + (\gamma\alpha)^2 \): \[ (\alpha\beta)^2 + (\beta\gamma)^2 + (\gamma\alpha)^2 = (\alpha\beta + \beta\gamma + \gamma\alpha)^2 - 2\alpha\beta\gamma(\alpha + \beta + \gamma) \] Substitute: \[ (\alpha\beta)^2 + (\beta\gamma)^2 + (\gamma\alpha)^2 = (3)^2 - 2(4)(2) = 9 - 16 = -7 \] So, \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -7 \).
Step 5: Compute \( \alpha^2\beta^2\gamma^2 \). 
Since \( \alpha\beta\gamma = 4 \), we have:
\[ \alpha^2\beta^2\gamma^2 = (\alpha\beta\gamma)^2 = (4)^2 = 16 \] Thus, the constant term in the new equation is \( -\alpha^2\beta^2\gamma^2 = -16 \). 
Step 6: Form the new cubic equation. 
Using the computed sums: 
Coefficient of \( x^2 \): \( -(\alpha^2 + \beta^2 + \gamma^2) = -(-2) = 2 \) 
Coefficient of \( x \): \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -7 \) 
Constant term: \( -\alpha^2\beta^2\gamma^2 = -16 \) The new cubic equation is: \[ x^3 + 2x^2 - 7x - 16 = 0 \] 
Step 7: Match with the options. 
The equation \( x^3 + 2x^2 - 7x - 16 = 0 \) matches option (2). 
Final Answer: 
\[ \boxed{x^3 + 2x^2 - 7x - 16 = 0} \]