Answer:
- Maximum of 2 positive real roots
- Exactly 1 negative real root
Step-by-step explanation:
Descartes' Rule of Signs tells us the maximum number of positive and negative real roots in a polynomial function.
Positive root case
[tex]\begin{aligned}p(x)&=(x)^3+5(x)^2-3(x)+7\\ &=+x^3+5x^2-3x+7\end{aligned}[/tex]
Check the signs of the coefficients. As there are 2 sign changes, the maximum possible number of positive roots is 2.
However, as some pairs of roots may be complex, count down by two's to find the complete list of the possible number of roots.
So there may be as many as 2 real roots or there might also be no real roots (zero).
Negative root case
[tex]\begin{aligned}p(-x)&=(-x)^3+5(-x)^2-3(-x)+7\\&=-x^3+5x^2+3x+7\end{aligned}[/tex]
Check the signs of the coefficients. As there is one sign change, the maximum possible number of negative roots is 1. We can't count down by two's since this would give us a negative number. Therefore, there is exactly 1 negative root.
Note: There is one real negative root of the given function at x ≈ -5.74 and two complex roots at x = 0.37 + 1.04i and x = 0.37 - 1.04i.
