The numbers in the bottom row alternate in sign. This indicates that −3 is a lower bound
for the real roots.
- Here is an outline of the process for finding the rational roots of the equation
3 x^5 − 3 x^2 + 2 x− 2 = 0
- All the coefficients, as well as the stand-alone constant, are integers, so we don’t have to
multiply the equation through by anything. - The integer factors m of the stand-alone constant are 2, 1, −2, and −1.
- The integer factors n of the leading coefficient are 3, 1, −3, and −1.
- All the possible ratios r=m/n are 2, 1, 2/3, 1/3, −2,−1,−2/3, and −1/3.
- We input rational numbers r of 2, 1, 2/3, 1/3, −2,−1,−2/3, and −1/3 to synthetic
division arrays, and see if we get a remainder of 0 for any of them. - We get a remainder of 0 only when r= 1. Therefore, x= 1 is the only rational root of
the equation.
Chapter 27
- Here are the two equations in their original forms:
3 x+y− 1 = 0
and
2 x^2 −y+ 1 = 0
We can morph these to obtain the following functions of x:
y=− 3 x+ 1
and
y= 2 x^2 + 1
When we mix the right sides of these equations, we obtain
− 3 x+ 1 = 2 x^2 + 1
which morphs into the quadratic equation
2 x^2 + 3 x= 0
Chapter 27 699