1. Use Descartes Rule of Signs to describe all possible roots:

X^4-3X^3+2X^2-X-7 = 0

2. Solve the equation:

5^(2X-1)=3^(X-3)

1. Use Descartes Rule of Signs to describe all possible roots:

X^4-3X^3+2X^2-X-7 = 0

Descartes rule of signs can be found on the net or in any algebra/pre-calc book. The number of positive REAL roots of f(x) is equal to the number of sign changes or less than that by an even number.

The number of negative REAL roots of f(-x) is equal to the number of sign changes or less than that by an even number.

Yours has 3 sign changes in f(x). Therefore, there are 3 or 1 positive roots.

Check f(-x).

There are 4 roots in all. Some may be complex.


2. Solve the equation:

5^(2X-1)=3^(X-3)

I will step through it using the log laws:

\frac{25^{x}}{5}=\frac{3^{x}}{27}

ln(\frac{25^{x}}{5})=ln(\frac{3^{x}}{27})

xln(25)-ln(5)=xln(3)-ln(27)

x(ln(25)-ln(3))=ln(5)-ln(27)

x(ln(\frac{25}{3}))=ln(\frac{5}{27})

x=\frac{ln(\frac{5}{27})}{ln(\frac{25}{3})}