College Algebra 9th txtbk

322 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS
3. Has at most n real zeros
4. Has at most n –1 turning points
5. Increases or decreases without bound as x S and as
x S
The left and right behavior of such a polynomial P(x) is determined
by its highest degree or leading term: As x S , both a n x n and
P(x) approach , with the sign depending on n and the sign of a n .
For any polynomial P(x) of degree n, we have the following
important results:
Division Algorithm
P(x) (x r)Q(x) R where the quotient Q(x) and remainder R
are unique; x – r is the divisor.
Remainder Theorem
P(r) R
Factor Theorem
x – r is a factor of P(x) if and only if R 0.
Zeros of Polynomials
P(x) has at most n zeros.
Synthetic division is an efficient method for dividing polynomials
by linear terms of the form x – r.
4-2 Real Zeros and Polynomial Inequalities
The following theorems are useful in locating and approximating
all real zeros of a polynomial P(x) of degree n 7 0 with real coefficients,
a n 7 0:
Upper and Lower Bound Theorem
1. Upper bound: A number r 7 0 is an upper bound for the real
zeros of P(x) if, when P(x) is divided by x – r using synthetic
division, all numbers in the quotient row, including the
remainder, are nonnegative.
2. Lower bound: A number r 6 0 is a lower bound for the real
zeros of P(x) if, when P(x) is divided by x – r using synthetic
division, all numbers in the quotient row, including the
remainder, alternate in sign.
Location Theorem
Suppose that a function f is continuous on an interval I that contains
numbers a and b. If f(a) and f (b) have opposite signs, then the
graph of f has at least one x intercept between a and b.
The bisection method uses the location theorem repeatedly to
approximate real zeros to any desired accuracy.
Polynomial inequalities can be solved by finding the zeros
and inspecting the graph of an appropriate polynomial with real
coefficients.
4-3 Complex Zeros and Rational Zeros of Polynomials
If P(x) is a polynomial of degree n 7 0 we have the following
important theorems:
Fundamental Theorem of Algebra
P(x) has at least one zero.
n Linear Factors Theorem
P(x) can be factored as a product of n linear factors.
If P(x) is factored as a product of linear factors, the number of
linear factors that have zero r is said to be the multiplicity of r.
Imaginary Zeros Theorem
Imaginary zeros of polynomials with real coefficients, if they exist,
occur in conjugate pairs.
Linear and Quadratic Factors Theorem
If P(x) has real coefficients, then P(x) can be factored as a product
of linear factors (with real coefficients) and quadratic factors (with
real coefficients and imaginary zeros).
Real Zeros and Polynomials of Odd Degree
If P(x) has odd degree and real coefficients, then the graph of P has
at least one x intercept.
Zeros of Even or Odd Multiplicity
Let P(x) have real coefficients:
1. If r is a real zero of P(x) of even multiplicity, then P(x) has a
turning point at r and does not change sign at r.
2. If r is a real zero of P(x) of odd multiplicity, then P(x) does not
have a turning point at r and changes sign at r.
Rational Zero Theorem
If the rational number b/c, in lowest terms, is a zero of the polynomial
P(x) a n x n a n1 x n1 # # # a 1 x a 0
with integer coefficients, then b must be an integer factor of a 0 and
c must be an integer factor of a n .
If P(x) (x r)Q(x), then Q(x) is called a reduced polynomial
for P(x).
4-4 Rational Functions and Inequalities
A function f is a rational function if it can be written in the form
where p(x) and q(x) are polynomials of degrees m and n, respectively.
The graph of a rational function f(x):
1. Is continuous with the exception of at most n real numbers
2. Has no sharp corners
3. Has at most m real zeros
f (x) p(x)
q(x)
4. Has at most m n – 1 turning points
a n 0
5. Has the same left and right behavior as the quotient of the
leading terms of p(x) and q(x)
The vertical line x a is a vertical asymptote for the graph
of y f(x) if f(x) S or f(x) S as x S a or as x S a . The
horizontal line y b is a horizontal asymptote for the graph of
y f(x) if f(x) S b as x S or as x S . The line y mx b
is an oblique asymptote if [ f(x) (mx b)] S 0 as x S or
as x S .