5-3-Ext

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Polynomials, Rational Expressions, and Closure
5-3-Ext
Lesson Presentation
Holt Algebra 2
Holt McDougal Algebra 2
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Objectives
Understand under which operations rational
expressions are closed.
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A set of numbers is closed, or has closure, under
a given operation if the result of the operation
on any two numbers in the set is also in the
set. For example, the set of real numbers is
closed under addition, because adding any two
real numbers results in another real number.
Likewise, the real numbers are closed under
subtraction, multiplication and division (by a
nonzero real number), because performing these
operations on two real numbers always yields
another real number. Polynomials are closed under
the same operations as integers. Rational
expressions are closed under addition,
subtraction, multiplication, and division by a
nonzero rational expression.
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Example 1 Determining Closure of the Set of
Integers Under Operations
Explain why the whole numbers are closed under
addition and multiplication but not under
subtraction and division.
Suppose that a and b are whole numbers. To find a
b, start at b on a number line and move right a
units. The result is another whole number.
Multiplication by a whole number is repeated
addition, so ab is also a whole number.
Subtraction counterexample 5 10 5, which is
not a whole number. Division counter-example 5
10 0.5, which is not a whole number.
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Check It Out! Example 1
Determine if the set of positive integers is
closed under addition, subtraction,
multiplication, and division. Explain.
The set of positive integers is only closed under
addition and multiplication. When adding positive
integers a and b, movement is to the right from
point a to point b by b units, which represents a
positive integer and each unit is an integer.
Since positive integers are closed under
addition, multiplication is also closed, as
multiplication of positive integers can be
rewritten as repeated addition. Positive integers
are not closed under subtraction.
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Check It Out! Example 1 continued
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Example 2 Determining Closure of the Set of
Rational Numbers Under Operations
Explain why irrational numbers are not closed
under multiplication.
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Check It Out! Example 2
Determine if the set of negative rational numbers
is closed under addition, subtraction,
multiplication, and division. Explain.
The set of negative rational numbers is closed
under addition. Since the addition of negative
integers is always negative, the addition of
negative rational numbers will result in a
negative rational number as well. The set of
negative rational numbers is not closed under
subtraction.
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Check It Out! Example 2 continued
Counterexample
. Since the result is a positive
rational number, negative rational numbers are
not closed under subtraction. The set of negative
rational numbers is not closed under
multiplication. Since the product of two negative
numbers is always a positive, then the product of
two rational numbers will never result in a
negative rational number. The same is true of
division of negative rational numbers. A division
of two negative numbers will result in a positive
number, so the set of negative rational numbers
is not closed under division.
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Example 3 Determining Closure of Polynomials
Explain why polynomials are not closed under
division.
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Check It Out! Example 3
Determine if the set of polynomials is closed
under subtraction with real-number coefficients.
The set of polynomials is closed under
subtraction. Since subtraction can be rewritten
as addition for the coefficients of like terms
and polynomials are closed under addition, they
are closed under subtraction for real-number
coefficients.
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Example 4 Determining Closure of Rational
Expressions
Show that rational expressions are closed under
subtraction.
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Example 4 continued
The product of two polynomials is a polynomial
and the difference of two polynomials is a
polynomial. Therefore, the numerator and
denominator of the result are polynomials, so the
result is a rational expression.
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Check It Out! Example 4
Determine if the set of rational numbers is
closed under multiplication.
Since the product of polynomials is closed under
multiplication, the rational expressions are
closed under multiplication.