Question

If \( f(x + h) = 0 \) represents the transformed equation of \[ f(x) = x^4 + 2x^3 - 19x^2 - 8x + 60 = 0 \] and this transformation removes the term containing \( x^3 \), then \( h \) is:

• A. \( -\frac{1}{2} \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( -1 \)

Hint:

For transformations that remove terms, use polynomial shifting \( x \to x + h \) and set unwanted coefficients to zero.

Answer 1

The Correct Option is A

Step 1: Condition for Eliminating \( x^3 \) Term Using the transformation \( x \to x + h \), the coefficient of \( x^3 \) must vanish. Step 2: Solve for \( h \) \[ \text{Coefficient of } x^3 \text{ in transformed equation} = 0 \] Solving, we obtain: \[ h = -\frac{1}{2} \] Thus, the correct answer is \( -\frac{1}{2} \).