Algebra Know-It-ALL

Part Three 523

Answer 26-10

We must do the following things, in the order shown below. The process can be tedious, but
it’s often more likely to produce useful results than tackling the equation “head-on” or hoping
for an intuitive breakthrough.

• Make sure that a 1 ,a 2 ,a 3 , ... an, and b are all integers. If they aren’t, multiply the equa-
  tion through by the smallest constant that will turn them all into integers.


• Find all the positive and negative integer factors of b. Call those factors m.


• Find all the positive and negative integer factors of an. Call those factors n.


• Write down all the possible ratios m/n. Call those ratios r.


• With synthetic division, check every r, one at a time, to see if we get a remainder
  of 0.


• If none of the numbers r produces a remainder of 0, then the original equation has no
  rational roots.


• If one or more of the ratios r produces a remainder of 0, then every one of those num-
  bers is a rational root of the equation.


• List all of the rational roots. Call them r 1 ,r 2 ,r 3 , and so on.


• Create binomials of the form (x−r 1 ), (x−r 2 ), (x−r 3 ), and so on. Each of these bino-
  mials is a factor of the original equation.


• If we’re lucky, this process will give us an equation in binomial to the nth form, or an
  equation in binomial factor form.


• If we’re less lucky, we’ll get one or more binomial factors and a quadratic factor. That
  factor can be set equal to 0, and then the quadratic formula can be used to find its
  roots. Neither of those roots will be rational. They might even be complex.


• If we’re unlucky, we’ll get one or more binomial factors and a cubic or higher-order
  polynomial factor. If we set the polynomial factor equal to 0, we can be sure that none
  of the roots associated with it are rational.

Chapter 27

Question 27-1

Suppose we’re confronted with a pair of equations in two variables, and one or both of the
equations is nonlinear. How can we solve these equations as a two-by-two system?

Answer 27-1

When we want to solve a general two-by-two system of equations, we can go through these
steps in order.

• Decide which variable to call independent, and which one to call dependent.


• Morph both equations so they express the dependent variable in terms of the indepen-
  dent variable.


• Mix the independent-variable parts of the equations to get an equation in one
  variable.


• Find the root(s) of that equation.