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Rational Zero Test
unio Mitsuma’s Web Site
Question: What’s
all about a shortcut in
the process of factoring and solving a polynomial equation
of a higher degree? |
Yours
Truly answers: The
shortcut mentioned in class refers to clever repetitive use
of the Factor Theorem in the course of solving a polynomial
equation. First, recall:
Example: Solve 6x4 – 29x3– 23x2 +
98x – 40 =
0.
Solution: Let f (x) = 6x4 – 29x3– 23x2 + 98x – 40. The factors of 40 are ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40 and the factors of 6 are ±1, ±2, ±3, ±6. We realize that there are too many possibilities to test. So, we will graph f (x) and try to guess some of the zeros of the function. By the graph, we see that x = –2 is a likely zero. Indeed, by synthetic division
 Namely, f (x) = 6x4 – 29x3– 23x2 + 98x – 40 = (x + 2)(6x3– 41x2 + 59x – 20). We still need to factor g(x) = 6x3– 41x2 + 59x – 20. By the graph again, we see that x = 5 is a likely zero. Indeed, by synthetic division  Namely, g(x) = 6x3– 41x2 + 59x – 20 = (x – 5)(6x2– 11x + 4). It
is easy to see that 6x2– 11x +
4 = (2x – 1)(3x – 4).
Consequently, f (x) = 6x4 – 29x3– 23x2 + 98x – 40 = (x + 2)(x – 5)(2x – 1)(3x – 4). Therefore, the solutions of f (x) = 0 are x = –2, 5, 1/2, and 4/3. Note 1: Graphing the function on TI-83 is an important step, so that we do not have to try all possible candidates for the zeros. Note 2: The two synthetic divisions can be combined into one, which further speeds up the computation. Hence, the phrase shortcut.  |
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Disclaimer • Updated September 4, 2008
Email: mitsuma@kutztown.edu
Phone: +1 (610) 462-WPJW
Phone: +1 (610) 462-WPJW