Question

The polynomial function f is given by f(x)= 2x³-3x²-23x 12. Which of the following is true about f (x)/x³?

1) f(x)/x³ is a polynomial function
2) f(x)/x³ is a rational function
3) f(x)/x³ is an irrational function
4) f(x)/x³ is an exponential function

Answer 1

When f(x) is divided by x³, the result is 2 - (3/x) - (23/x²) + (12/x³), which is a rational function. A rational function is defined as a ratio of two polynomials, making option 2) the correct answer.

Step-by-step explanation:

The question concerns the type of function represented by f(x)/x³ where f(x) = 2x³ - 3x² - 23x + 12. To determine this, one must understand the definitions of polynomial, rational, irrational, and exponential functions.

A polynomial function is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. When you divide the polynomial f(x) by x³, each term of f(x) is divided by x³, resulting in the expression 2 - (3/x) - (23/x²) + (12/x³).

Since this new expression involves division by a variable, it is no longer a polynomial. Instead, it fits the definition of a rational function, which is a ratio of two polynomials. Therefore, option 2) is correct; f(x)/x³ is a rational function.