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

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

### Relevant Questions

Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}$$
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={4}{x}^{{{3}}}+{4}{x}^{{{2}}}-{x}-{1}$$
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={4}{x}^{{{3}}}-{7}{x}+{3}$$
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={4}{x}^{{{4}}}-{37}{x}^{{{2}}}+{9}$$
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$P(x) = 3x^{4} - 5x^{3} + x^{2} - 3x +5$$
$$Q(x) = (((3x - 5)x + 1)x 3)x + 5$$
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial
R(x) =x^{5} - 2x^{4} + 3x^{3} - 2x^{2} + 3x + 4\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={6}{x}^{{{4}}}-{23}{x}^{{{3}}}-{13}{x}^{{{2}}}+{32}{x}+{16}$$
$$\displaystyle{P}{\left({x}\right)}={6}{x}^{{{4}}}-{7}{x}^{{{3}}}-{12}{x}^{{{2}}}+{3}{x}+{2}$$
$$\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{14}{x}^{{{4}}}-{14}{x}^{{{3}}}+{36}{x}^{{{2}}}+{43}{x}+{10}$$
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}$$
$$\displaystyle{P}{\left({x}\right)}={12}{x}^{{{3}}}-{25}{x}^{{{2}}}+{x}+{2}$$