Polynomials,
Rational
Polynomials,
Rational
5-3-Ext
5-3-Ext
Expressions,
and
Closure
Expressions,
and
Closure
Lesson Presentation
HoltMcDougal
Algebra 2Algebra 2
Holt
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Objectives
Understand under which operations
rational expressions are closed.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
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.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
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 counterexample: 5 ÷ 10 = 0.5, which is not a whole
number.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
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.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Check It Out! Example 1 continued
Counterexample: for a = 5 and b = 7, a – b =
5 – 7 = –2, which is a negative integer.
Positive integers are not closed under division.
Counterexample: for a = 3 and b = 8, a = 3
b
8
, which is not a positive integer.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Example 2: Determining Closure of the Set of
Rational Numbers Under Operations
Explain why irrational numbers are not closed
under multiplication.
A counterexample is √3 • √3 . The result is √9 =
3, which is not an irrational number.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
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.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Check It Out! Example 2 continued
Counterexample: a  c   1    3    1  3   2  3  1 .
b
d
2
 4
2
4
4
4
4
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.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Example 3: Determining Closure of Polynomials
Explain why polynomials are not closed under
division.
A counterexample is (x2 + 1) ÷ (x - 5). The result is
x2 + 1 , which is a rational expression, not a
x+4
polynomial.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
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.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Example 4: Determining Closure of Rational
Expressions
Show that rational expressions are closed under
subtraction.
Let f(x), g(x), p(x), and q(x) be polynomial
f(x)
p(x)
expressions. Then
and
represent rational
g(x)
q(x)
expressions, where g(x) ≠ 0 and q(x) ≠ 0. To subtract
the functions, use a common denominator.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Example 4 continued
f(x)
p(x)
f(x) . q(x) p(x) . g(x)
+
=
g(x)
q(x)
g(x) q(x) q(x) g(x)
f(x).q (x) – p(x).g(x)
=
g(x).q(x)
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.
Holt McDougal Algebra 2
5-3-Ext
Polynomials, Rational
Expressions, and Closure
Check It Out! Example 4
Determine if the set of rational numbers is
closed under multiplication.
The set of rational expressions is closed under
multiplication. Let f(x), g(x), p(x), and q(x) be
polynomial expressions and g(x) ≠ 0 and q(x) ≠ 0.
f(x) . p(x)
f(x).p(x)
g(x) q(x) = g(x).q(x)
Since the product of polynomials is closed under
multiplication, the rational expressions are closed
under multiplication.
Holt McDougal Algebra 2