Guided Example #1 - Polynomial Equations

is 2. By Descarte's rule, the number of positive real roots is either 2 or 0. Since ${\displaystyle P(-x)=2{(-x)}^{15}+3{(-x)}^8-4{(-x)}^7+3{(-x)}^4+6{(-x)}^3+{(-x)}^2+1}$
${\displaystyle =-2x^{15}+3x^8+4x^7+3x^4-6x^3+x^2+1}$
there are three variations of sign in ${\displaystyle P(-x)}$. So there are either 1 or 3 negative real roots.

The equation must have 15 roots, because it is a fifteenth- degree polynomial equation. Since there must be 15 roots and there are 0, or two positive and 1 or 3 negative, the other 10,12, or 14 roots must be non-real.

The number of variations of sign in ${\displaystyle P\left(x\right)=2x^{15}+3x^8-4x^7+3x^4+6x^3+x^2+1}$

is 2. By Descarte's rule, the number of positive real roots is either 2 or 0. Since ${\displaystyle P(-x)=2{(-x)}^{15}+3{(-x)}^8-4{(-x)}^7+3{(-x)}^4+6{(-x)}^3+{(-x)}^2+1}$
${\displaystyle =-2x^{15}+3x^8+4x^7+3x^4-6x^3+x^2+1}$
there are three variations of sign in ${\displaystyle P(-x)}$. So there are either 1 or 3 negative real roots.

The equation must have 15 roots, because it is a fifteenth- degree polynomial equation. Since there must be 15 roots and there are 0, or two positive and 1 or 3 negative, the other 10,12, or 14 roots must be non-real.