- There are only two integer factors n of the leading coefficient: 1 and −1.
- All the possible ratios m/n are the same as the integers m: 24, 12, 8, 6, 4, 3, 2, and 1, along with
all their negatives.
- Now let’s cheat a little. Imagine that we’ve narrowed down the interval by doing synthetic division
repeatedly, finding a smallest upper bound of 5 and a greatest lower bound of −4. That leaves us
with rational numbers r of 4, 3, 2, 1, −1,−2, and −3 to check as possible roots.
- We input 4, 3, 2, 1, −1,−2, and −3 to synthetic division arrays, and see if we get a remainder of
0 for any of them.
- We get a remainder of 0 when r= 4, r= 2, r=−1, or r=−3. Now we know that every one of those
numbers is a rational root of the equation.
- We have found four rational roots for a fourth-degree equation. There are no more rational roots to
find! In fact, these are all the roots of any sort. Remember: A polynomial equation can never have more
roots than its degree. That includes not only the rational roots, but the irrational and complex roots.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. C. The solutions in the appendix may
not represent the only way a problem can be figured out. If you think you can solve a particu-
lar problem in a quicker or better way than you see there, by all means try it!
- Rewrite each of the following equations in binomial to the nth form.
(a) (x^2 + 6 x+ 9)^2 = 0
(b) (x^2 − 4 x+ 4)^3 = 0
(c) (16x^2 − 24 x+ 9)^4 = 0
- What are the real roots for each of the equations stated in Prob. 1? What is the
multiplicity in each case?
- Rewrite each of the following equations in binomial factor form.
(a) (x^2 − 3 x+ 2)^2 = 0
(b) (− 3 x^2 − 5 x+ 2)^5 = 0
(c) (4x^2 − 9)^3 = 0
- What are the real roots for each of the equations stated in Prob. 3? What is the
multiplicity in each case?
- State the real roots of the following equation. Also state the real solution set X and the
multiplicity of each root. What is the degree of the equation?
(x− 3/2)^2 (2x− 7)^2 (7x)^3 (− 3 x+ 5)^5 = 0
- State the real roots of the following equation. Also state the real solution set X and the
multiplicity of each root. What is the degree of the equation?
(x+ 4)(2x− 8)^2 (x/3+ 12)^3 = 0
Practice Exercises 445
