Chapter 4 – Polynomials and Rational Functions

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Chapter 4 Polynomials and Rational Functions
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4.1 Polynomial Functions
Def A polynomial in one variable, x, is an
expression of the form . The coefficients
a0, a1, a2, , an represent complex numbers (real
or imaginary), a0 is not zero, and n represents a
nonnegative integer. Def The degree of a
polynomial in one variable is the greatest
exponent of its variable. Def If a function f
is a polynomial in one variable, then f is a
polynomial function. Def If p(x) represents a
polynomial, then p(x) 0 is called a polynomial
equation. Def A root of the equation is a
value of x for which the value of the polynomial
p(x) is 0. It is also called a zero.
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Ex Determine if each expression is a polynomial
in one variable. If so, state the
degree. a. b. c. Ex Determine
whether 3 is a root of
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What is an imaginary number? What is a
complex number? Fundamental Theorem of
Algebra Every polynomial equation with degree
greater than zero has at least one root in the
set of complex numbers
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Corollary to the Fundamental Theorem of
Algebra Every polynomial p(x) of degree n can be
written as the product of a constant k and n
linear factors. Thus, a polynomial equation of
degree n has exactly n complex roots, namely r1,
r2, r3, , rn. Relationship with degree and
roots
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Ex State the number of complex roots of the
equation x3 2x2 8x 0. Then find the roots
and graph the related polynomial
function. Ex Write the polynomial
equation of least degree with roots -3 and 2i.
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4.2 Quadratic Equations and Inequalities
Ex Solve each equation by completing the
square. a. b. Quadrat
ic Formula The roots of a quadratic equation of
the form ax2 bx c 0 with a not equal to
zero are given by the following formula.
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Ex Solve 4x2 8x 3 0 using the quadratic
formula. Then graph the related function.
Discriminant b2 4ac gt 0 two distinct
real roots b2 4ac 0 exactly one real root
(double root) b2 4ac lt 0 no real roots
(imaginary roots)
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Ex Determine the discriminant of x2 6x 13
0. Use the quadratic formula to find the roots.
Then graph the related function.
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Ex Graph y gt x2 8x - 20
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4.3 The Remainder and Factor Theorems
The Remainder Theorem If a polynomial p(x) is
divided by x r, the remainder is a constant,
p(r), and where q(x) is a polynomial with
degree one less than the degree of
p(x). Example Let p(x) x3 3x2 2x 8.
Show that the value of p(-2) is the remainder
when p(x) is divided by x 2.
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Ex Use synthetic division to divide m5 3m2
20 by m 2. The Factor Theorem The
binomial x r is a factor of the polynomial p(x)
if and only if p(r) 0. Ex Let p(x) x3
4x2 7x 10. Determine if x 5 is a factor of
p(x).
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4.4 The Rational Root Theorem Rational Root
Theorem Let
represent a polynomial equation of
degree n with integral coefficients. If a
rational number p/q, where p and q have no common
factors, is a root of the equation, then p is a
factor of an and q is a factor of
a0. Example Possible values for p Possible
values for q Possible rational roots
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Integral Root Theorem Let
represent a polynomial
equation that has leading coefficients of 1,
integral coefficients, and . Any
rational roots of this equation must be integral
factors of an. Ex Find the roots of x3 6x2
10x 3 0. Descartes Rule of
Signs Suppose p(x) is a polynomial whose terms
are arranged in descending powers of the
variable. Then the number of positive real zeros
of p(x) is the same as the number of changes in
sign of the coefficients of the terms, or is less
than this by an even number. The number of
negative real zeros of p(x) is the same as the
number of changes in sign of the coefficients of
the terms of p(-x), or is less than this by an
even number.
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Ex State the number of possible complex zeros,
the number of positive real zeros, and the number
of possible negative real zeros for h(x) x4
2x3 7x2 4x -15. Ex Find the zeros of
M(x) x4 4x3 3x2 4x 4. Then graph the
function.
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4.5 Locating the Zeros of a Function The
Location Principle Suppose y f(x) represents a
polynomial function. If a and b are two numbers
with f(a) negative and f(b) positive, the
function has at least one real zero between a and
b. Ex Determine between which consecutive
integers the real zeros of f(x) x3 2x2 3x
-5 are located.
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Ex Approximate to the nearest tenth the real
zeros of f(x) x4 3x3 2x2 3x 5. Then
sketch the graph of the function, given that the
relative maximum is at (0.4, -4.3) and the
relative minima are at (-0.7, -6.8) and (2.5,
-17.8).
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Upper Bound Theorem Suppose c is a positive
integer and p(x) is divided by x c. If the
resulting quotient and remainder have no change
in sign, then p(x) has no real zeros greater than
c. Thus, c is an upper bound of zeros of
p(x). Ex Find a lower bound of the zeros of
f(x) x4 3x3 2x2 3x 5.
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4.6 Rational Equations and Partial
Fractions Ex Solve Ex Solve
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Decompose into
partial fractions Solve
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Solve
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4.7 Radical Equations and Inequalities Solve
Solve
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Ex Solve Ex Solve