1. Polynomials, Rational Expressions, and Closure
5-3-Ext
Lesson Presentation
Holt McDougal Algebra 2
Holt Algebra 2

2. Objectives Understand under which operations
rational expressions are closed.

3. 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.

4. 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.

5. 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.

6. 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, = , which is not a positive integer. a b 3 8

7. 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.

8. 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.

9. 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.

10. Example 3: Determining Closure of Polynomials
Explain why polynomials are not closed under division.
x2 + 1
x + 4
A counterexample is (x2 + 1) ÷ (x - 5). The result is
, which is a rational expression, not a
polynomial.

11. 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.

12. 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
expressions. Then and represent rational
expressions, where g(x) ≠ 0 and q(x) ≠ 0. To subtract the functions, use a common denominator.
f(x)
g(x)
p(x)
q(x)

13. Example 4 continued
f(x)
g(x)
f(x).q (x) – p(x).g(x)
g(x).q(x)
p(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.

14. 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)
g(x).q(x) =
f(x)
g(x)
p(x)
q(x) .
Since the product of polynomials is closed under multiplication, the rational expressions are closed under multiplication.