UkusakazaL

2020-11-14

We need find:

The real zeros of polynomial function form an arithmetic sequence

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

The real zeros of polynomial function form an arithmetic sequence

lamusesamuset

Skilled2020-11-15Added 93 answers

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$g(x)\ne 0.$

Calculation:

The given polynomial unction form an arithmetic sequence is

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

Here, the constant is 0.

The above equation can be rewritten as

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

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

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

Where, p(x) and g(x) are polynomial functions, with

Calculation:

The given polynomial unction form an arithmetic sequence is

Here, the constant is 0.

The above equation can be rewritten as

The possibilities for

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