Algebra Know-It-ALL

698 Worked-Out Solutions to Exercises: Chapters 21 to 29

• 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 2, 1, −2, and −1.


• All the possible ratios r=m/n are 2, 1, 1/2, −2,−1, and −1/2.


• We input rational numbers r of 2, 1, 1/2, −2,−1, and −1/2 to synthetic division arrays,
  and see if we get a remainder of 0 for any of them.


• We don’t get a remainder of 0 when we input any of the above values of r to the syn-
  thetic division array. Therefore, the equation has no rational roots.

9. For reference, the polynomial equation is

3 x^5 − 3 x^2 + 2 x− 2 = 0

Let’s use the same method as we did in solution to Prob. 7. The largest absolute value of any

coefficient or constant is 3, so we can try 3 for the upper bound and −3 for the lower bound.

If either or both of these fail, we can try values farther from 0. The coefficients and constant,

in order of decreasing powers of x, are 3, 0, 0, −3, 2, and −2. (The coefficients of x^4 and x^3

are both equal to 0.) Here’s the synthetic division array for the “test root” 3:

(^3300) − 3 2 − 2

When we go through the synthetic division process, we end up with
3300 − 3 2 − 2
9 27 81 234 708
3 9 27 78 236 706
None of the numbers in the bottom row are negative. This tells us that 3 is an upper
bound for the real roots. To try the “test root” −3, we set up the array
3300 − 3 2 − 2

The synthetic division process leads us to
− 3 300 − 3 2 − 2
− 9 27 − 81 252 − 762
(^3) − 9 27 − 84 254 − 764