Algebra 2AAlgebra 2A

7.17.1

September 19, 2007September 19, 2007

ObjectivesObjectives

Simplify rational expressionsSimplify rational expressions Identify rational functions.Identify rational functions. Simplify rational functions.Simplify rational functions. Graph rational functions.Graph rational functions.

A little vocabularyA little vocabulary

Expression: a combination of numbers and Expression: a combination of numbers and mathematical operations that does not contain mathematical operations that does not contain an equal signan equal sign

Function: an equation for which each input has Function: an equation for which each input has one outputone output

Rational: the ratio of two integers (for Rational: the ratio of two integers (for numbers), the ratio of two polynomials (for numbers), the ratio of two polynomials (for expressions or functions)expressions or functions)

Simplify: non-reducible formSimplify: non-reducible form

What makes an expression What makes an expression undefined?undefined?

x x .. x-5x-5When is a fraction undefined? When is a fraction undefined? What must the denominator equal?What must the denominator equal?

x-5=0x-5=0x=5x=5This expression is undefined when x=5 becauseThis expression is undefined when x=5 because5 5 is undefined. is undefined.00DIVISION BY ZERO MAKES AN EXPRESSION DIVISION BY ZERO MAKES AN EXPRESSION

UNDEFINED.UNDEFINED.

How do you evaluate a rational How do you evaluate a rational expression?expression?

3x 3x ..2x-52x-5Evaluate the expression above for x=4.Evaluate the expression above for x=4.Plug 4 in for x.Plug 4 in for x.3(4)3(4)2(4)-52(4)-5In the calculator, key in “3 * 4 / (2 * 4 -5)” and press In the calculator, key in “3 * 4 / (2 * 4 -5)” and press

enter.enter.The answer is 4.The answer is 4.

Simplifying rational expressionsSimplifying rational expressionsFundamental principle of rational expressionsFundamental principle of rational expressionsPP == PRPRQQ == QRQR where Qwhere Q≠0 and R≠0≠0 and R≠0

So, when faced with a rational expression, look for a common factor So, when faced with a rational expression, look for a common factor in order to simplify.in order to simplify.

12121515What is the common factor?What is the common factor?

3 * 43 * 43 * 53 * 5

12 12 == 4 415 515 5

More simplifyingMore simplifying

4x4x22yy 3x-63x-6

12xy12xy22 xx22-4-4

x+2x+2 5x5x22-5-5

x+3x+3 xx22-4x-5-4x-5

Something to help you factorSomething to help you factor

a-b = -(-a+b) = -(b-a) = -1(b-a)a-b = -(-a+b) = -(b-a) = -1(b-a)

a-ba-b == -(b-a) -(b-a) = -1= -1

b-ab-a b-ab-a

How does this work?How does this work?

2x-42x-4

4-x4-x22

Steps for simplifying any rational Steps for simplifying any rational expressionexpression

1.1. Completely factor both the numerator and Completely factor both the numerator and the denominator.the denominator.

2.2. Divide the numerator and denominator by all Divide the numerator and denominator by all common factors.common factors.

3.3. The resulting expression will be in the The resulting expression will be in the simplest form (in the lowest terms).simplest form (in the lowest terms).

Rational functionsRational functions

f(x) = f(x) = PP

QQ

Where P and Q are polynomials are Q Where P and Q are polynomials are Q ≠ 0≠ 0

Which of the following are rational functions?Which of the following are rational functions?

f(x) = 3xf(x) = 3x22-2x+5-2x+5

f(x) = f(x) = 3x3x22-2x+5-2x+5

2x-52x-5

f(x) = 3xf(x) = 3x33+ 3√x+ 3√x

Simplifying rational functionsSimplifying rational functions

f(x)=f(x)= xx22 +2x +1 +2x +1 x+1x+1

What makes the denominator equal zero? This represents a hole What makes the denominator equal zero? This represents a hole in the graph.in the graph.

Now simplify:Now simplify:f(x) = f(x) = (x+1) (x+1(x+1) (x+1)) == x+1x+1

x+1x+1Plug the “hole” into the simplified expression to find its ordered Plug the “hole” into the simplified expression to find its ordered

pairpair-1 +1 =0-1 +1 =0

The ordered pair is (-1, 0)The ordered pair is (-1, 0)

Graphing a rational functionGraphing a rational function

f(x)=f(x)= xx22 +2x +1 +2x +1

x+1x+1

Simplified: x+1Simplified: x+1

Where is the hole? Mark the Where is the hole? Mark the hole on the graph with an hole on the graph with an open circle.open circle.