Final answer:
To simplify the function, divide each term in the numerator by x, the denominator, to get f(x) = x² + 7x + 1 - 5x⁻¹. Subtract exponents for terms with the same base, while constants and terms without exponents are divided normally.
Step-by-step explanation:
To simplify the function f(x) = (x³ + 7x² + x - 5) / x, we can divide each term of the numerator by the denominator. The function simplifies to:
x³ / x = x²
7x² / x = 7x
x / x = 1
-5 / x = -5x⁻¹ (which is -5/x)
The simplified result is f(x) = x² + 7x + 1 - 5x⁻¹.
According to the rules for division of exponentials, when you divide powers with the same base, you subtract the exponents. For example, x³ / x is x raised to the power of (3 - 1), which is x². The constant terms and variables without exponents are also divided normally.
This method is a way of simplifying algebraic expressions, and it is important to always check that your final answer is reasonable and that all terms have been properly simplified.