Algebra Know-It-ALL

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