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

UkusakazaL 2020-11-14 Answered
We need find:
The real zeros of polynomial function form an arithmetic sequence
f(x)=x44x34x2+16x.
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Expert Answer

lamusesamuset
Answered 2020-11-15 Author has 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)=pxq(x) is a rational function.
Where, p(x) and g(x) are polynomial functions, with g(x)0.
Calculation:
The given polynomial unction form an arithmetic sequence is
f(x)=x44x34x2+16x.
Here, the constant is 0.
The above equation can be rewritten as
f(x)=x(x34x24x+16)
The possibilities for pqare±1,±2,±4,and±8. Factoring the term (x34x24x+16), we get (x34x24x+16)=(x+2)(x26x+8) Factoring the term (x26x+8), we get (x26x+8)=(x2)(x4) Combining all the terms, we get (x34x24x+16)=(x+2)(x2)(x4)
f(x)=x(x34x24x+16)=x(x+2)(x2)(x4) Thus, the real zeros are -2, 0, 2, and 4
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