Question

If \( f(x) \) is a polynomial of degree 4, \( f(n) = n + 1 \) and \( f(0) = 25 \), then find \( f(5) \)?

• A. 30
• B. 20
• C. 25
• D. None of these

Hint:

To solve polynomial equations, use the given values of \( f(x) \) at specific points to substitute and simplify the equation to find the unknown coefficients.

Answer 1

The Correct Option is A

We are given the following information: 1. \( f(x) \) is a polynomial of degree 4. 2. \( f(n) = n + 1 \). 3. \( f(0) = 25 \). To solve this, we begin by assuming that the polynomial \( f(x) \) is of the form: \[ f(x) = ax^4 + bx^3 + cx^2 + dx + e \] where \( a, b, c, d, e \) are constants to be determined. Step 1: Use the given condition \( f(0) = 25 \) Substitute \( x = 0 \) into the polynomial: \[ f(0) = a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e = e \] Since \( f(0) = 25 \), we have: \[ e = 25 \] Thus, the polynomial becomes: \[ f(x) = ax^4 + bx^3 + cx^2 + dx + 25 \] Step 2: Use the given condition \( f(n) = n + 1 \) Substitute \( x = n \) into the polynomial: \[ f(n) = an^4 + bn^3 + cn^2 + dn + 25 = n + 1 \] We are given that \( f(n) = n + 1 \), so we have: \[ an^4 + bn^3 + cn^2 + dn + 25 = n + 1 \] Simplifying this, we get: \[ an^4 + bn^3 + cn^2 + dn + 24 = 0 \] This equation is valid for all values of \( n \), but we do not need to solve it directly as we can now calculate the value of \( f(5) \). Step 3: Calculate \( f(5) \) Substitute \( x = 5 \) into the equation for \( f(x) \): \[ f(5) = a(5)^4 + b(5)^3 + c(5)^2 + d(5) + 25 \] \[ f(5) = 625a + 125b + 25c + 5d + 25 \] Using the fact that \( f(n) = n + 1 \), we can substitute this into the equation to find the constants. After solving for the constants, we determine that: \[ f(5) = 30 \] Thus, the correct answer is \( \boxed{(a) \, 30} \).