8 Intro to Rational Expressions PDF

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Math 043:

Intro to Rational Expressions

If we form the quotient of two polynomials, the result is called a rational expression. Some examples of rational expressions are xy 2 x3  1 3 x2  x  2 2 ( x  y) x x2  5 Rational expressions are described in the same manner as rational numbers. In the first expression above, x3 + 1 is the numerator, and x is the denominator. When the numerator and denominator of a rational expression contain no common factors (except 1 and -1), we say that the rational expression is reduced to lowest terms, or simplified. The polynomial in the denominator of a rational expression cannot be equal to 0 because division by zero is x3 1 , x cannot take on the value 0. The domain of the undefined. For example, for the expression x variable x is x ≠ 0. NOTE: If the denominator of a rational expression is never equal to 0, then there are no values for which the rational expression is undefined, and the domain is all real numbers. Examples: If possible, evaluate the expression for the given value of the variable. 3 , 1. x  4 x = 5

x 3 , 2. 3  x x = -2

3.

4y , y 2  9 y = -3

Examples: Find all numbers for which the rational expression is undefined.

1.

3 x 4

x 2  8x  12 2 4. x  5 x  6

.

8x 2. 3 x  7

5.

x−7 x 2−49

x 4 2 3. x  9

6.

8 x2 +x +1 2 x 2−9 x +7

Math 043:

Intro to Rational Expressions

A rational expression is simplified to lowest terms by factoring the numerator and denominator completely and dividing any common factors.

Basic Principle of Rational Expressions:

P ⋅R P = where Q∧R are nonzero Q⋅R Q

Examples: Simplify each expression.

3 x 6 1. 4 x  8

( x  3)( x  8) 2. ( x  2)( x  3)

 2x  10 3. x 5

7 x 4. x  7

x 2  3x  4 2 5. x  9x  8

x2  x 2 6. x  9x

x 2  16 2 7. x  3x  28

3 x 2  9 x  30 x2  25 8.

x 2  10x  24 6 x 9.

10.

4 x 2+23 x−6 12 x 2+13 x− 4

11.

x−2 3 x −11 x + 10 2

Math 043:

Intro to Rational Expressions

Insect Population Suppose that an insect population in thousands per acre is modeled by P=

5 x+2 x+1

where x ≥ 0 is time in months.

a) Complete the table by finding P for each given value of x. Round to 3 decimal places.

X (months)

0

12

P (thousands)

b) What was the initial insect population?

c) Interpret the results in your completed table.

36

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