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CAAP Math Homework

1. (a) Let be a binary operation defined for real numbers ; this means that for any real
numbers x and y , we have some definition of xy.

We say that the operation is commutative if for all real numbers x and y ,

xy = yx.

We say that the operation is associative if for all real numbers x, y, and z,

x(yz) = (xy)z.

In the following questions, if the answer is "no", then give a counterexample. a pair
of numbers that shows that the operation is not commutative, or three numbers
that show that the operation is not associative.

(b) Is addition of real numbers commutative?
(c) Is multiplication of real numbers commutative?
(d) Is subtraction of real numbers commutative?
(e) Is division of nonzero real numbers commutative?
(f) Is addition of real numbers associative?
(g) Is multiplication of real numbers associative?
(h) Is subtraction of real numbers associative?
(i) Is division of nonzero real numbers associative?
(j) Define the operation on the nonzero integers byIs commutative? Is it associative?
(k) Define the operation on the integers by Is commutative? Is
it associative?
(l) Is it true that xy = 0 if and only if x = 0 or y = 0? If not, what is the condition on
x and y given by xy = 0? (Give an exact characterization of when this happens,
not just some examples.)

2. Factor the polynomial x^2 + 1000002x + 2000000.

3. Try to reduce each fraction to lowest terms .
(a) Convert the decimal to a fraction.
(b) Convert the decimal to a fraction.
(c) Convert the decimal to a fraction.

4. What is

5. Use long division of polynomials to compute the quotient and the remainder of

6. Starting with the curve defined by we apply the transformation
that shifts to the right by 1 unit. Which of the following is the equation of the resulting
curve ?

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