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# We need find: The real zeros of polynomial function form an arithmetic sequence f(x) = x^{4} - 4x^{3} - 4x^{2} + 16x. # We need find: The real zeros of polynomial function form an arithmetic sequence f(x) = x^{4} - 4x^{3} - 4x^{2} + 16x.

Question
Polynomial arithmetic asked 2020-11-14
We need find:
The real zeros of polynomial function form an arithmetic sequence
$$f(x) = x^{4} - 4x^{3} - 4x^{2} + 16x.$$

## Answers (1) 2020-11-15
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

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