Question

Let \( H(x) = 3x^4 + 6x^3 - 2x^2 + 1 \) and \( g(x) \) be a linear polynomial. If \[ \frac{H(x)}{(x-1)(x+1)(x-2)} = f(x) + \frac{g(x)}{(x-1)(x+1)(x-2)}, \] then find \( H(-1) + 2H(2) - 3H(1) \).

• A. \(f(-1) + 2f(2) - 3f(1)\)
• B. \(H(-1) + f(2) + g(3)\)
• C. \(g(-1) + 2g(2) - 3g(1)\)
• D. \(H(1) + 2f(2) - g(1)\)

Hint:

Use factorization and evaluation at roots to simplify expressions.

Answer 1

The Correct Option is C

Step 1: Use the given decomposition
\[ \frac{H(x)}{(x-1)(x+1)(x-2)} = f(x) + \frac{g(x)}{(x-1)(x+1)(x-2)} \] Step 2: Multiply both sides by denominator
\[ H(x) = f(x)(x-1)(x+1)(x-2) + g(x) \] Step 3: Evaluate at \(x = -1, 2, 1\)
At \(x=-1\): \[ H(-1) = g(-1) \] At \(x=2\): \[ H(2) = g(2) \] At \(x=1\): \[ H(1) = g(1) \] Step 4: Compute
\[ H(-1) + 2H(2) - 3H(1) = g(-1) + 2g(2) - 3g(1) \]