Pregunta
Use Descartes’ rule of signs to determine the total number of real zeros and the number of positive and negative real zeros. (Hint: First factor out
to its lowest power.)
Part 1 of 3
Number of possible positive real zeros:
N
Part 1 of 3
Number of possible positive real zeros:
N
Ask by Graham Cook. in the United States
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The number of possible positive real zeros is 2 or 0.
Solución
To apply Descartes’ rule of signs, we first need to factor out
from the polynomial
.
- Factor out
:
Now we will analyze the polynomial
to determine the number of positive real zeros.
- Count the sign changes for positive real zeros:
- The coefficients of
are: . - The signs of the coefficients are:
.
- The coefficients of
Now, we will list the signs:
-
(for ) -
(for ) -
(for ) -
(for ) -
(for )
The sign changes occur as follows:
- From
to : 1 change - From
to : 0 changes - From
to : 1 change
Thus, there are a total of 2 sign changes.
According to Descartes’ rule of signs, the number of positive real zeros is equal to the number of sign changes or less than that by an even number. Therefore, the possible number of positive real zeros is:
- 2 or 0.
Thus, the number of possible positive real zeros is
or
.
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Bonus Knowledge
To apply Descartes’ rule of signs, we first factor out the lowest power of
from the polynomial
. Factoring out
, we rewrite the polynomial as
.
Next, we consider the remaining polynomial
for positive real zeros. To do this, we look for sign changes in the coefficients:
- Positive:
(no sign change) - Negative:
(sign change) - Negative:
(sign change) - Positive:
(sign change)
Counting the sign changes, we find three sign changes in total, which indicates there could be 3, 1, or no positive real zeros.
Now, let’s sum it up: There are 3 possible positive real zeros from our analysis using Descartes’ rule of signs!

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