Concept:

A rational expression is a fraction that is quotient of two polynomials. A rational function is defined by the two polynomial functions.

A function f of the form,\(f(x)= p\frac{x}{q}(x)\) is a rational function.

Where, p(x) and g(x) are polynomial functions, with \(g(x) \neq 0.\)

Calculation:

The given polynomial unction form an arithmetic sequence is

\(f(x) = x^{4} - 4x^{3} - 4x^{2} + 16x.\)

Here, the constant is 0.

The above equation can be rewritten as

\(f(x) = x(x^{3} - 4x^{2} - 4x + 16)\)

The possibilities for \(\frac{p}{q} are \pm 1, \pm 2, \pm 4, and \pm 8.\) Factoring the term \((x^{3} - 4x^{2} -4x + 16)\), we get \((x^{3}- 4x^{2} -4x + 16)=(x+2)(x^{2}-6x+8)\) Factoring the term \((x^{2}-6x+8)\), we get \((x^{2}-6x+8)=(x-2)(x-4)\) Combining all the terms, we get \((x^{3}- 4x^{2} -4x + 16)=(x+2)(x-2)(x-4)\)

\(f(x) = x(x^{3} - 4x^{2} - 4x + 16) = x(x + 2)(x - 2)(x - 4)\) Thus, the real zeros are -2, 0, 2, and 4

A rational expression is a fraction that is quotient of two polynomials. A rational function is defined by the two polynomial functions.

A function f of the form,\(f(x)= p\frac{x}{q}(x)\) is a rational function.

Where, p(x) and g(x) are polynomial functions, with \(g(x) \neq 0.\)

Calculation:

The given polynomial unction form an arithmetic sequence is

\(f(x) = x^{4} - 4x^{3} - 4x^{2} + 16x.\)

Here, the constant is 0.

The above equation can be rewritten as

\(f(x) = x(x^{3} - 4x^{2} - 4x + 16)\)

The possibilities for \(\frac{p}{q} are \pm 1, \pm 2, \pm 4, and \pm 8.\) Factoring the term \((x^{3} - 4x^{2} -4x + 16)\), we get \((x^{3}- 4x^{2} -4x + 16)=(x+2)(x^{2}-6x+8)\) Factoring the term \((x^{2}-6x+8)\), we get \((x^{2}-6x+8)=(x-2)(x-4)\) Combining all the terms, we get \((x^{3}- 4x^{2} -4x + 16)=(x+2)(x-2)(x-4)\)

\(f(x) = x(x^{3} - 4x^{2} - 4x + 16) = x(x + 2)(x - 2)(x - 4)\) Thus, the real zeros are -2, 0, 2, and 4