Algebra Know-It-ALL

444 Polynomial Equations in Real Numbers

• With synthetic division, check every r to see if it is a root of the original polynomial
  equation. If you find a root, you’ll get a remainder of 0 at the end of the synthetic divi-
  sion process.


• If none of the numbers r produces a remainder of 0, then the original polynomial
  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 found after carrying out the preceding steps. 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 you’re lucky, you’ll end up with an equation in binomial to the nth form, or an equa-
  tion in binomial factor form.


• If you’re less lucky, you’ll end up with 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 you’re unlucky, you’ll be stuck with one or more binomial factors and a cubic or
  higher-degree factor. If you set that factor equal to 0 to form a polynomial equation,
  you’ll know that none of the roots of that equation are rational. Some might even be
  complex. You can try to solve it, but you should not expect the task to be easy.

Are you confused?

At this point, you must wonder, “Suppose we’re left with a cubic or higher-degree polynomial as one of the

factors, and its associated equation has some irrational roots. What can we do to find those roots?” The best

answer is, “We can use a computer program to generate an approximate graph of the function produced by

the polynomial equation, see how many times that graph crosses the x axis, and then use the computer to

approximate the zeros of that function.” This method will not allow us to find non-real roots.

Here’s a challenge!

Use the above-described procedure to find all the rational roots of the polynomial equation we contrived

earlier in this chapter, and for which we know the smallest upper bound is 5 and the greatest lower bound

is−4. Once again, that equation is

x^4 − 2 x^3 − 13 x^2 + 14 x+ 24 = 0

Solution

Here is an outline of the process. You might want to work out the arithmetic, particularly the synthetic

division problems, to verify.

• 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 positive and negative integer factors of the stand-alone constant, 24, are all the integers m that
  divide 24 without remainders. These numbers are 24, 12, 8, 6, 4, 3, 2, and 1, along with all their
  negatives.