Key Concepts and Formulas

• Polynomial Root Theorem: If $r$ is a root of a polynomial $P(x)$, then $P(r) = 0$.
• Polynomial Factorization: If $r_1, r_2, ..., r_n$ are roots of a polynomial $P(x)$ of degree $n$, then $P(x) = C(x-r_1)(x-r_2)...(x-r_n)$, where $C$ is a constant.
• Degree of a Polynomial: The highest power of the variable in the polynomial.

Step-by-Step Solution

Step 1: Formulate a New Polynomial We are given that $f(x)$ is a polynomial of degree 3, and $f(k) = -\frac{2}{k}$ for $k = 2, 3, 4, 5$. We want to create a new polynomial that has roots at $x = 2, 3, 4, 5$. To do this, we manipulate the given equation. Multiplying both sides of $f(k) = -\frac{2}{k}$ by $k$, we get: $k f(k) = -2$ Adding 2 to both sides, we obtain: $k f(k) + 2 = 0$ This tells us that the expression $x f(x) + 2$ equals zero when $x = 2, 3, 4, 5$. Therefore, we can define a new polynomial $P(x) = x f(x) + 2$, which has roots at $2, 3, 4, 5$.

Step 2: Determine the Degree of P(x) Since $f(x)$ is a polynomial of degree 3, we can write it in the general form: $f(x) = ax^3 + bx^2 + cx + d$ where $a \neq 0$. Now we find the expression for the polynomial $P(x)$: $P(x) = x f(x) + 2 = x(ax^3 + bx^2 + cx + d) + 2$ $P(x) = ax^4 + bx^3 + cx^2 + dx + 2$ The highest power of $x$ in $P(x)$ is $x^4$, and since $a \neq 0$, the degree of $P(x)$ is 4.

Step 3: Construct the Polynomial P(x) using its roots We know that $P(x)$ is a polynomial of degree 4 with roots $2, 3, 4, 5$. Therefore, we can write it in the form: $P(x) = C(x-2)(x-3)(x-4)(x-5)$ where $C$ is a constant. Substituting $P(x) = x f(x) + 2$, we get: $x f(x) + 2 = C(x-2)(x-3)(x-4)(x-5) \quad (*)$

Step 4: Determine the Constant C To find the value of $C$, we need to substitute a value for $x$ (other than 2, 3, 4, or 5) into equation $(*)$ and solve for $C$. The easiest value to use is $x=0$. Substituting $x=0$ into the equation: $0 \cdot f(0) + 2 = C(0-2)(0-3)(0-4)(0-5)$ $2 = C(-2)(-3)(-4)(-5)$ $2 = C(120)$ $C = \frac{2}{120} = \frac{1}{60}$ Thus, the polynomial equation is: $x f(x) + 2 = \frac{1}{60}(x-2)(x-3)(x-4)(x-5)$

Step 5: Calculate 10f(10) We are asked to find the value of $52 - 10f(10)$. To do this, we need to find the value of $10f(10)$. Substitute $x=10$ into the equation: $10 f(10) + 2 = \frac{1}{60}(10-2)(10-3)(10-4)(10-5)$ $10 f(10) + 2 = \frac{1}{60}(8)(7)(6)(5)$ $10 f(10) + 2 = \frac{1}{60}(1680)$ $10 f(10) + 2 = 28$ $10 f(10) = 28 - 2$ $10 f(10) = 26$

Step 6: Calculate 52 - 10f(10) Now we substitute the value of $10f(10)$ into the expression: $52 - 10f(10) = 52 - 26$ $52 - 10f(10) = 26$

Common Mistakes & Tips

• Be careful when calculating the constant $C$. A simple arithmetic error can lead to an incorrect answer.
• Always check the degree of the polynomials involved to ensure consistency.
• Choosing $x=0$ to find the constant $C$ simplifies the calculation.

Summary

By constructing a new polynomial $P(x) = xf(x) + 2$ with roots at $2, 3, 4, 5$, we were able to determine its general form and find the constant factor. Then, by substituting $x=10$ into the equation, we found the value of $10f(10)$, which allowed us to calculate $52 - 10f(10)$. The final result is 26.

Final Answer The final answer is \boxed{26}.