Question

The polynomial equation of degree 5 whose roots are the roots of the equation $$ x^5 - 3x^4 + 11x^2 - 12x + 4 = 0 $$ each increased by 2 is

• A. \( x^5 - 13x^4 + 63x^3 - 135x^2 - 108x = 0 \)
• B. \( x^5 - 13x^4 + 63x^3 + 135x^2 + 108x = 0 \)
• C. \( x^5 - 13x^4 + 63x^3 - 135x^2 + 108x = 0 \)
• D. \( x^5 - 13x^4 - 63x^3 - 135x^2 - 108x = 0 \)

Hint:

For shifted roots, substitute \( x = y + k \) or \( x - k \), and expand. Look for patterns or match coefficients.

Answer 1

The Correct Option is C

Let the original equation be \( f(x) \), and we want \( g(x) = f(x - 2) \) Perform the substitution: \[ f(x - 2) = (x - 2)^5 - 3(x - 2)^4 + 11(x - 2)^2 - 12(x - 2) + 4 \] Expand each term and combine like terms (or use binomial expansion and verify options). Upon simplification, we get: \[ x^5 - 13x^4 + 63x^3 - 135x^2 + 108x \]