Question

The cubic equation whose roots are the squares of the roots of the equation \( 12x^3 - 20x^2 + x + 3 = 0 \) is:

• A. \( x^3 + 376x^2 - 121x - 9 = 0 \)
• B. \( 144x^3 - 400x^2 + 121x + 98 = 0 \)
• C. \( 144x^3 - 376x^2 + 121x - 9 = 0 \)
• D. \( x^3 + 400x^2 - 121x - 98 = 0 \)

Hint:

Remember to apply the same steps and use Vieta's formulas for cubic equations where you need to relate roots and their powers.

Answer 1

The Correct Option is C

To find the cubic equation whose roots are the squares of the roots of the given equation \( 12x^3 - 20x^2 + x + 3 = 0 \), we proceed as follows: Step 1: Let the roots of the original equation Let the roots of the original equation \( 12x^3 - 20x^2 + x + 3 = 0 \) be \( r, s, t \). Then: \[ r + s + t = \frac{20}{12} = \frac{5}{3}, \quad rs + rt + st = \frac{1}{12}, \quad rst = -\frac{3}{12} = -\frac{1}{4}. \] Step 2: Find the roots of the new equation The roots of the new equation are the squares of the roots of the original equation, i.e., \( r^2, s^2, t^2 \). We need to find the sums and products of these new roots. #Sum of the new roots: \[ r^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + rt + st) = \left(\frac{5}{3}\right)^2 - 2\left(\frac{1}{12}\right) = \frac{25}{9} - \frac{1}{6} = \frac{50}{18} - \frac{3}{18} = \frac{47}{18}. \] #Sum of the products of the new roots: \[ r^2s^2 + r^2t^2 + s^2t^2 = (rs + rt + st)^2 - 2rst(r + s + t) = \left(\frac{1}{12}\right)^2 - 2\left(-\frac{1}{4}\right)\left(\frac{5}{3}\right) = \frac{1}{144} + \frac{5}{6} = \frac{1}{144} + \frac{120}{144} = \frac{121}{144}. \] #Product of the new roots: \[ r^2s^2t^2 = (rst)^2 = \left(-\frac{1}{4}\right)^2 = \frac{1}{16}. \] Step 3: Construct the new equation The new cubic equation with roots \( r^2, s^2, t^2 \) is: \[ x^3 - (r^2 + s^2 + t^2)x^2 + (r^2s^2 + r^2t^2 + s^2t^2)x - r^2s^2t^2 = 0. \] Substituting the values from Step 2: \[ x^3 - \frac{47}{18}x^2 + \frac{121}{144}x - \frac{1}{16} = 0. \] To eliminate fractions, multiply through by \( 144 \): \[ 144x^3 - 376x^2 + 121x - 9 = 0. \] Step 4: Match with the options The equation \( 144x^3 - 376x^2 + 121x - 9 = 0 \) matches option (3). Final Answer: \[ \boxed{3} \]