f(x) = log {((x^2)+ab)/(x(a+b))} is it a polynomial function

The given function is:

\[
f(x) = \log\left(\frac{x^2 + ab}{x(a + b)}\right)
\]

### Determining if it's a polynomial function:
A **polynomial function** is an expression that consists only of powers of the variable \(x\) with non-negative integer exponents, and it has the general form:

\[
f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0
\]

#### Key observations:
1. **Logarithm:** The presence of a logarithmic function means \(f(x)\) cannot be a polynomial function. Polynomial functions do not involve logarithms, as they only consist of sums and products of powers of \(x\).
   
2. **Rational expression inside the logarithm:** Even without the logarithm, the expression inside is a **rational function** (a ratio of two polynomials), not a polynomial. The term inside is \(\frac{x^2 + ab}{x(a + b)}\), which simplifies to \(\frac{x^2 + ab}{x}\), but still, this wouldn't be a polynomial due to the division.

### Conclusion:
The function \( f(x) = \log\left(\frac{x^2 + ab}{x(a + b)}\right) \) is **not** a polynomial function. The logarithmic nature and the division of polynomials disqualify it from being a polynomial.

Would you like further details or explanations on logarithmic or polynomial functions?

Here are some related questions:
1. What are the main differences between polynomial and rational functions?
2. How do you simplify logarithmic expressions like the one in this function?
3. Can a function involving logarithms ever be a polynomial?
4. What is the general form of a logarithmic function?
5. How does the domain of a function involving logarithms affect its behavior?

**Tip:** Polynomial functions are smooth and continuous everywhere, unlike logarithmic functions, which have restricted domains.

Explore whether the function f(x) = log((x^2 + ab) / (x(a + b))) is a polynomial. Understand key concepts like polynomial functions, logarithms, and rational expressions. Detailed explanation with formulas and theorems for students in Grades 10-12.

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