Answer:
We conclude that:
- 2 or 0 positive real roots
- 2 or 0 negative real roots
Step-by-step explanation:
Descartes Rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is:
- Either equal to the number of sign differences between consecutive nonzero coefficients,
- Or is less than it by an even number.
Given the function
[tex]f\left(x\right)=x^4-2x^3-4x^2+2x+3[/tex]
So, the coefficients are 1, −2, −4, 2, 3
As can be seen, there are 2 changes.
This means that there are 2 or 0 positive real roots.
To find the number of negative real roots, substitute x with -x in the given polynomial:
[tex]x^4-2x^3-4x^2+2x+3[/tex] becomes [tex]x^4+2x^3-4x^2-2x+3[/tex]
The coefficients are 1, 2, −4, −2, 3
As can be seen, there are 2 changes.
This means that there are 2 or 0 negative real roots.
Therefore, we conclude that:
- 2 or 0 positive real roots
- 2 or 0 negative real roots
