Respuesta :
f(-x)=3(-x)^4-5(-x)^3+5(-x)^2+5(-x)+2
=3x^4+5x^3+5x^2-5x+2
=+ + + - +
there are two changes of sign
so at the most two negative real roots
=3x^4+5x^3+5x^2-5x+2
=+ + + - +
there are two changes of sign
so at the most two negative real roots
Answer:
The correct option is D) Two or zero.
Step-by-step explanation:
According to Descartes' rule of signs: It is used to determine the number of real zeros of a polynomial function.
Determine the number of positive and negative real zeros for the given function ;
[tex]f(x)=3x^{4}-5x^{3}+5x^{2}+5x+2[/tex]
Given function is arranged in descending powers of the variable.
Here are the coefficients of our variable in f(x):
+3 - 5 +5 +5 +2
Our variables goes from positive(3) to negative(-5) to positive(5) to positive(5) to to positive(2).
Between our first and second we have our first change, then between our second and third we have our second change but between the third to fourth coefficients there are no change in signs and similarly in fourth to fifth coefficients there are no change in signs.
Descartes´ rule of signs tells us that the we then have exactly 2 real positive zeros
In order to find the number of negative zeros we find f(-x) and count the number of changes in sign for the coefficients:
[tex]f(-x)=3(-x)^{4}-5(-x)^{3}+5(-x)^{2}+5(-x)+2[/tex]
[tex]f(-x)=3x^{4}+5x^{3}+5x^{2}-5x+2[/tex]
Here are the coefficients of our variable in f(-x):
+3 +5 +5 - 5 +2
There is two change in sign of variables
According to Descartes´ rule of signs ,we have Two negative real root.
Therefore, the correct option is D) Two or zero.
