Descartes’ Rule of Signs and Rational Root Theorem

Descartes’ Rule of Signs and Rational Root Theorem
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Descartes’ Rule of Signs and Rational Root Theorem
For questions 246 to 248, discuss the positive or negative nature of the roots of
the given polynomial equation. Refer to the following rule, as needed.
Descartes’ rule of signs: If the polynomial equation p(x) = 0 has real
coefficients, its maximum number of positive real roots either equals the number
of sign variations in p(x) or is less than that by an even number; and its
maximum number of negative real roots either equals the number of sign
variations in p(–x) or is less than that by an even number. Note: When the terms
of p(x) are written in descending (or ascending) order of powers of x, a sign
variation occurs if the coefficients of two consecutive terms have opposite signs
(with missing terms being ignored).
246. 6x
4 7x
3 – 9x
2 – 7x 3 = 0
247. x
3 – 1 = 0
248. x
5 4x
4 – 4x
3 – 16x
2 3x 12 = 0
249. List the possible rational roots for 2x
3 x
2 – 13x 6 = 0.
250. Find the real roots for 2x
3 x
2 – 13x 6 = 0.
Rational Functions
Descartes’ Rule of Signs and Rational Root Theorem
In questions 251 to 256, for the given rational function (a) state the domain, (b)
find the zeros, (c) find the x-intercepts, and (d) find the y-intercepts. Refer to the
following guidelines, as needed.
A rational function f is defined as , where p(x) and q(x) are
polynomials. Its domain is a subset of R, excluding all values of x for which q(x)
= 0; and its range is a subset of R. When is in simplified form, the
zeros occur at x values for which p(x) = 0; the x-intercepts occur at real values
for which p(x) = 0; and if 0 is in the domain, the y-intercept is f(0).
251.
252.
253.
254.
255.
256.
Vertical, Horizontal, and Oblique Asymptotes and Holes in
Graphs of Rational Functions
For questions 257 to 261, (a) determine vertical asymptote(s) and (b) hole(s).
Refer to the following definitions, as needed.
Suppose is in simplified form, then
a. The vertical line with equation x = a is a vertical asymptote of the graph of f
if f(x) either increases or decreases without bound, as x approaches a from
the left or right. To determine vertical asymptotes, set the denominator
equal to 0 and solve. See the following graph.
b. Upon inspection, if p(x) and q(x) have a common factor, (x – h), that will
divide out completely from the denominator when f(x) is simplified, then
the graph will have a “hole” at (h, f (h)), where f (h) is calculated after f(x)
is simplified. See the following graph.
257.
258.
259.
260.
261.
For questions 262 to 266, determine the horizontal or oblique asymptote. Refer
to the following guidelines, as needed.
Suppose is in simplified form, then a horizontal or oblique (slanted)
line is an asymptote of the graph of f if f(x) approaches the line as x approaches
positive or negative infinity. See the following graph for an example of a
horizontal asymptote.
To determine horizontal or oblique asymptotes for , compare the
degrees of the numerator and denominator polynomials. Note: The graph may
have either a horizontal or an oblique asymptote—one or the other—but not
both.
• If the numerator’s degree is less than the denominator’s degree, then the xaxis is a horizontal asymptote.
• If the numerator’s degree equals the denominator’s degree, then the graph
has a horizontal asymptote at , where an
is the leading coefficient of
p(x) and bn
is the leading coefficient of q(x).
Descartes’ Rule of Signs and Rational Root Theorem
• If the numerator’s degree exceeds the denominator’s degree by exactly 1,
the graph has an oblique asymptote. To find the equation of the oblique
asymptote, use division to rewrite as quotient plus .
The line with equation y = quotient is the oblique asymptote.
• If the numerator’s degree exceeds the denominator’s degree by more than 1,
the graph has neither a horizontal nor an oblique asymptote.
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