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Rational Algebraic Expressions:

Exploring Activity
Introduction to rational algebraic expressions

Goal: To prove the knowledge related to rational numbers and fraction basic operations.

Instructions:
1. Make an individual reflexion concern on the following questions:
a) How a rational number is defined?

In mathematics, a rational number is any number that can be expressed asthe quotient or
fraction a/b of two integers, with thedenominator b not equal to zero.

b) Why dividing between zero, the answer is undefined?

The reason why dividing by zero is undefined has to do with algebra. It has nothing to do with calculus or taking limits. (Even in advanced math, if you work with more abstract "numbers" which have no concept of limits or "approaching zero," division byzero is still undefined.)

Think about what the equation x/y = z means. It means that x = y * z. So what would it mean to say that x/0 = z? It would mean that x = 0 * z = 0. If x is nonzero, then this is impossible. That's why there's no way to define x/0 when x is nonzero.

If x=0, then you have a different problem. Now the equation 0 = 0 * z has too many solutions; any value of z will satisfyit, so there's no sensible way to pick a value for 0/0. For example, if you tried to say 0/0 = 1, here's one problem that would cause. It would break the rules of algebra, because you would get
1 = 0/0 = (2*0)/0 = 2*(0/0) = 2*1 = 2
if you tried to follows the normal rules. So the only reasonable thing to do is to say 0/0 is undefined.

c) Which are homogeneous fractions? List some examples.Multivariate functions that are "homogeneous" of some degree are often used in economic theory. A function is homogeneous of degree k if, when each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by tk. For example, a function is homogeneous of degree 1 if, when all its arguments are multiplied by any number t > 0, the value of the function ismultiplied by the same number t.
Example
For the function  f (x1, x2) = Ax1ax2b with domain {(x1, x2): x1 ≥ 0 and x2 ≥ 0} we have
 f (tx1, tx2) = A(tx1)a(tx2)b = Ata+bx1ax2b = ta+b f (x1, x2),
so that  f  is homogeneous of degree a + b.

d) How are homogeneous fractions added and subtracted?

The first rule of addition is that only homogeneous fractions (with the same denominator) can beadded.
If the fractions have the same denominator, add or subtract the numerators and put the same denominator.

e) Which are not homogeneous fractions’ List some examples.

f) How are not homogeneous fractions added and subtracted.
g) Which is the rule to multiply fractions? Is this multiplication rule only applied to homogeneous fractions?
h) Which is the rule to divide fractions? Isdivision rules only applied to homogeneous fractions?

2. Find the following key term definitions by consulting your text book
a) Polynomial
b) Degree of a polynomial.
c) List examples related to grade 1, grade 1, grade 3 and grade 15 polynomials.
d) List three examples related to grade zero polynomial.
e) Rational algebraic expression.
f) List three rational algebraic expression examples.
g)Why in a rational algebraic expression, the polynomial at the denominator cannot be zero?
h) In a rational algebraic expression, can the denominator be a whole number?

3. Elaborate a document in which the answer to all question above must be included, conclusion, and the main idea related to all this work.


Knowledge Acquired Activity:

Evaluating Rational Algebraic Expressions:
Goal: Toevaluate Rational Algebraic Expressions by substitution.

Instructions:
1. Individually evaluate the following rational algebraic expressions given the indicated value:

When x = 0; x = 3; x = -2

When x = 0; x = 4; x = -1

When x = 0; x = 2; x = 4

2. As a group activity led by the professor, share comments about the answers obtained, mainly respond to these following...
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