Question

If \( f(x) = ax^2 + bx + c \) is an even function and \( g(x) = px^3 + qx^2 + rx \) is an odd function, and if \( h(x) = f(x) + g(x) \) and \( h(-2) = 0 \), then \( 8p + 4q + 2r = \)?

• A. \( 4a + 3b + 2c \)
• B. \( a + b + c \)
• C. \( 4a + 2b + c \)
• D. \( 8a + 4b + 2c \)

Hint:

For problems involving even and odd functions, leverage the properties of these functions to simplify equations and solve for unknowns efficiently. Equating values derived from these properties can directly lead to the solution.

Answer 1

The Correct Option is C

Since \(f(x)\) is an even function, it satisfies \(f(-x) = f(x)\). For \(g(x)\), being an odd function, it satisfies \(g(-x) = -g(x)\). Given \(h(-2) = 0\), substituting \(-2\) into the function \(h(x)\), we have: \[ h(-2) = f(-2) + g(-2) = f(2) - g(2) = 0 \] This implies \(f(2) = g(2)\). To express \(f(2)\) and \(g(2)\) in terms of their coefficients, we use: \[ f(2) = 4a + 2b + c, \quad g(2) = 8p + 4q + 2r \] Given \(f(2) = g(2)\), it follows that: \[ 4a + 2b + c = 8p + 4q + 2r \] Therefore, \(8p + 4q + 2r\) equals \(4a + 2b + c\), matching the correct answer option.