Lesson 6 Closed Minded Practice Understanding

Learning Focus

Support or challenge claims about different types of numbers and the result of adding, subtracting, multiplying, and dividing.

Support or challenge claims about the result of adding, subtracting, multiplying, and dividing polynomials.

What are the similarities and differences between the arithmetic of integers, rational numbers, real numbers, complex numbers, and polynomials?

What does the formal definition of a polynomial mean?

What does it mean for a set of numbers to be closed under an operation?

Open Up the Math: Launch, Explore, Discuss

Now that we have compared operations on polynomials with operations on whole numbers and started thinking about new number sets including irrational and complex numbers, it’s time to generalize the results.

Maybe you have noticed in the past that when you add two even numbers, the answer you get is always an even number. Mathematically, we say that the set of even numbers is closed under addition. Mathematicians are interested in results like this because it helps us to understand how numbers or functions of a particular type behave with the various operations.

1.

You can try it yourself: Is the set of odd numbers closed under addition? In other words if you add two odd numbers together will you get an odd number? Explain.

If you find any two odd numbers that have an even product, then you would say that odd numbers are not closed under multiplication. Even if you have several examples that support the claim, if you can find one counterexample that contradicts the claim, then the claim is false.

Consider the following claims and determine whether they are true or false. If a claim is true, give a reason with at least two examples that illustrate the claim. If the claim is false, give a reason with one counterexample that proves the claim to be false.

This graphic will help you to think about the relationship between different sets of numbers, including the complex numbers that we have found as solutions to quadratic equations.

Do the following for each of the following claims:

Determine if the claim is true or false.
If you decide that the claim is true, make a general argument that explains why it is always true or create at least three examples to support the claim.
If you decide that the claim is false, find a counterexample to prove the claim is false.

2.

The set of integers is closed under addition.

3.

The set of irrational numbers is closed under addition.

4.

The sum of a rational number and an irrational number is irrational.

5.

The set of whole numbers is closed under subtraction.

6.

The set of rational numbers is closed under subtraction.

7.

The set of integers is closed under multiplication.

8.

The set of integers is closed under division.

9.

The set of rational numbers is closed under multiplication.

10.

The set of irrational numbers is closed under multiplication.

11.

The set of complex numbers is closed under multiplication.

Part 2: The Arithmetic of Polynomials

To evaluate similar claims about polynomials, we must be very clear on the definition of a polynomial.

A polynomial function has the form:

$f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$

where $a_{n}, a_{n - 1}, \dots a_{1}, a_{0}$ are real numbers and $n$ is a nonnegative integer. In other words, a polynomial is the sum of one or more monomials with real coefficients and nonnegative integer exponents. The degree of the polynomial function is the highest value for $n$ where $a_{n}$ is not equal to $0$ .

12.

The following examples and non-examples will help you to see the important implications of the definition of a polynomial function. For each pair, determine what is different between the example of a polynomial and the non-example.

These are polynomials:	These are not polynomials:
$f (x) = x^{3}$	$g (x) = 3^{x}$
How are $f (x)$ and $g (x)$ different?
$f (x) = 2 x^{2} + 5 x - 12$	$g (x) = \frac{2 x^{2}}{x^{2} - 3 x + 2}$
How are $f (x)$ and $g (x)$ different?
$f (x) = - x^{3} + 3 x^{2} - 2 x - 7$	$g (x) = x^{3} + 3 x^{2} - 2 x + 10 x^{- 1} - 7$
How are $f (x)$ and $g (x)$ different?
$f (x) = \frac{1}{2} x$	$g (x) = \frac{1}{2 x}$
How are $f (x)$ and $g (x)$ different?
$f (x) = x^{2}$	$g (x) = x^{\frac{1}{2}}$
How are $f (x)$ and $g (x)$ different?

13.

Based on the definition and the examples above, how can you tell if a function is a polynomial function?

Now we will consider claims about polynomials. Do the following for each of the following claims:

Determine if the claim is true or false.
If you decide that the claim is true, create at least two examples to support the claim.
If you decide that the claim is false, find a counterexample to prove the claim to be false.

14.

The sum of a quadratic polynomial function and a linear polynomial function is a cubic polynomial function.

15.

The sum of a linear polynomial function and an exponential function is a polynomial function.

16.

A cubic polynomial function subtracted from a cubic polynomial function is a cubic polynomial function.

17.

A cubic polynomial function divided by a linear polynomial function is a quadratic polynomial.

18.

The set of polynomial functions is closed under addition.

19.

The set of polynomial functions is closed under subtraction.

20.

The set of polynomial functions is closed under multiplication.

21.

The set of polynomial functions is closed under division.

Ready for More?

Write two claims of your own about polynomials and use examples to demonstrate that they are true.

Claim #1:

Claim #2:

Takeaways

Closure Properties:

Adding Notation, Vocabulary, and Conventions

The definition of a polynomial implies:

Vocabulary

closure
polynomial function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined claims about the closure of sets of numbers and classes of functions under the operations of addition, subtraction, multiplication, and division. An example of such a claim is: The set of whole numbers is closed under division. A counterexample that shows this claim to be false is: $2 \div 7 = \frac{2}{7}$ . Since $\frac{2}{7}$ is a rational number, this example shows that dividing two whole numbers does not always result in a whole number.

Retrieval

1.

a.

Find the zeros for $f (x) = - 3 x^{2} - 3 x + 6$ .

b.

Mark the solutions as points on the graph of $f (x)$ .

2.

Find the product. $(x - \sqrt{5}) (x + \sqrt{5})$