Solution:
Given:
[tex]P(x)=x^9+2x^5-7x^3-x+2[/tex]a) The degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients.
The degree of P(x) is 9.
b) The polynomial P(x) will have 9 linear factors. It has degree 9, hence will have 9 roots or solutions. Each root can be made as a linear factor.
Therefore, the function P(x) will have 9 linear factors.
c) P(x) has 3 real solutions.
d) P(x) has 6 complex solutions.
e) Descartes rule of signs
Descartes's rule of signs is a rule for determining the maximum number of positive real number solutions (roots) of a polynomial equation in one variable based on the number of times that the signs of its real number coefficients change when the terms are arranged in the canonical order.
f) x = 2 is not a solution to P(x)
Applying the factor theorem,
[tex]\begin{gathered} \text{If a is a root to a polynomial,} \\ \text{then, f(a)=0} \end{gathered}[/tex]Thus,
[tex]\begin{gathered} P(x)=x^9+2x^5-7x^3-x+2 \\ P(2)=2^9+2(2^5)-7(2^3)-2+2 \\ P(2)=512+2(32)-7(8)-2+2 \\ P(2)=512+64-56-2+2 \\ P(2)=520 \\ \\ S\text{ ince P(2)}\ne0,\text{ then x = 2 is not a solution.} \end{gathered}[/tex]Therefore, x = 2 is not a solution to P(x)
