21(x * x *x) ÷ 7 (x*x)

Otcq

Aim 2-1: How do we define and simplify rational expressions?

HWk read 2-1 p 67# 1-10

Objective: SWBAT Simplify a Rational Expression.

Objectives:1.SWBAT to define a rational expression.2.SWBAT to simplify a rational expression to its simplest form.3. SWBAT to find prohibited values for variables in denominators that would make an expression undefined.4. SWBAT to +, -, * and ÷ with rational expressions and then to factor and to simplify rational expressions.

Review: Laws of Exponents

amnMultiplying Powers: *

m na a

Laws of Exponents

am n

amn

ambm

Dividing Powers:

Power of a Power:

Power of a Product:

m

n

a

a

nma

mab

Laws of Exponents

1

an

an

Negative Exponents:

na

1n

a

Laws of Exponents n

a

b

Power of a Quotient:

Power of Zero: 0a

n

n

a

b

1

nb

a

n

n

a

b

Examples using the laws of exponents:

1. 2ab2 3a4b2c 6a5b4c

2. 6x2y3 xyz 6x3y4z

3. x12

x4 x8

Definition: A Rational Expression is the quotient of two polynomials with the denominator not equal to zero.

25

2

m+4 8x 2 5, , , x

m-4 4x 5

x x

y x

EXCLUDED VALUES IN DENOMINATORS

Any value of x that makes the denominator = 0 is prohibited from the expression.

Why?

Because an expression is undefined when its denominator is equal to 0.

For test show algebra: x+ 2 = 0 -2 -2 x = - 2

If x= -1, then -1 + 1 = 0.

The expression is undefined when a variable value makes the denominator equal to 0

Answer: 7

x 1

RECALL:

Rational number = any number that may be expressed as a quotient of two integers with no 0 denominator.

Now we have:

Rational expression = any expression that may be stated as a quotient of two polynomials with no 0 denominator.

Remember, denominators cannot = 0.

Now, lets go through the steps to simplify a rational expression.

Examples of rational expressions

2

4 8 4 7, ,

3 3 5 9

x y

x x y y

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Would you like to review factoring of trinomials?

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Are the polynomials in ax2 + bx + c form?

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Yes, each polynomial is in ax2 + bx + c form?So for x2 + 6x + 5 we need:___ + ___ = b___ * ___ = c

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Yes, each polynomial is in ax2 + bx + c form?So for x2 + 6x + 5 we need:___ + ___ = 6___ * ___ = 5

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Yes, each polynomial is in ax2 + bx + c form?So for x2 + 6x + 5 we need:_5__ + _1__ = 6_5__ * _1__ = 5 so our numerator is

(x+5)(x+1) x2 - 25

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Can we factor the denominator?Do you recognize DOTS?

x2 + 6x + 5 = (x+5)(x+1) x2 - 25 x2 - 25

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Can we factor the denominator?Do you recognize DOTS? a2 – b2 = (a-b)(a+b)

x2 + 6x + 5 = (x+5)(x+1) x2 - 25 x2 - 25

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Can we factor the denominator?Do you recognize DOTS? a2 – b2 = (a-b)(a+b)So our denominator of x2 – 25 = (x - )(x + )

x2 + 6x + 5 = (x+5)(x+1) x2 - 25 x2 - 25

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Can we factor the denominator?Do you recognize DOTS? a2 – b2 = (a-b)(a+b)So our denominator of x2 – 25 = (x -5)(x +5 )

x2 + 6x + 5 =(x+5)(x+1) = (x+5)(x+1) x2 - 25 x2 – 25 (x-5)(x+5)

Writing a rational expression in simplest form.Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

Can we cancel like binomials as like factors?Yes!Our final answer is = x+1 x-5

x2 + 6x + 5 =(x+5)(x+1) = (x+5)(x+1) x2 - 25 x2 – 25 (x-5)(x+5)

Simplify: 7x 7

x2 1

Step 1: Factor the numerator and the denominator completely looking for common factors.

7x 7 7(x 1)

x2 1 (x 1)(x 1)Next

7x 7

x2 1

7(x 1)

(x 1)(x 1)

What is the common factor?x 1

Step 2: Divide the numerator and denominator by the common factor.

7(x 1)

(x 1)(x 1)

7(x 1)

(x 1)(x 1)

1

1

Step 3: Cancel and simplify.

Answer: 7

x 1

How do I find the values that make an expression undefined?

Completely factor the original denominator.

Ex: 2ab(a 2)(b 3)

3ab(a2 4)How do we determine when this is undefined? Cross out Numerator. Factor the denominator

Ex: 2ab(a 2)(b 3)

3ab(a2 4)

Ex: 2ab(a 2)(b 3)

3ab(a2 4)

3ab(a2 4) 3ab(a 2)(a 2)

Set factors = 0 one at a time and solve. The expression is undefined when: a= 0, 2, and -2 and b= 0. End test #2.

Factor the denominator

On the Regents EXAM, “Simplest form” means all common factors have been canceled. So,

Step 1: Factor both numerator and denominator completely.

Step 2: Cancel common factors and simplify.

2

2

2

2

3 2 22

8 x x) = 1

8 6 6

( 1) 3 3) =

8 =

48 6

2 3 ( 3) =

( 2)3 2

2 =

4

1 ( 1) ( 1) =

1( 1) 2 2

) Already in lowest terms

( 1)) = 1 1

( 11 ) 1

xA

x

x x xB

x x x

C

xD x

x x

x

x x x

x

xx

x x

y

y

x x x x x

x

Lets go through another example. Put this expression in simplest form.

3a3 a4

2a3 6a2

3a3 a4

2a3 6a2 a3 (3 a)

2a2 (a 3)

Factor out the GCF

Next

3

22 ( 3)

(3 )a

a a

a

3 factored is 1( 3)a a

cancel like factors3

2

1 ( 3)

2 ( 3)

a a

a a

1

1

3

2

1( 3)

2 ( 3)

a

a a

a

KEY TRICK

3

2

1

2

a

a

2cancel out the like factor a

1

2

a1

a

answer

For what values will the original expression be undefined? Go back to prior slide and set factors = 0.

Now try to do some on your own.

Put these in their simplest form.

2

2

3 2

3 2

5 61)

9

5 102)

6 16

x x

x

x x

x x x

Also find the values that make each expression undefined? Time permitting start hwk.

Multiplying Rational Expressions. With rational expressions, we always factor first and then cancel common factors in numerators and denominators before we multiply.

Ex: 4a2

5ab3 3bc

12a3 4 a a 3 bc

5 a b b b 12 a a a

11 1 1 1

1 1 1 1

c

5b2 a2

Let’s do another one.

Ex: x3 3x2

x2 5x 6

x2 10x 9

x2 6x 27Step #1: Factor the numerator and the denominator.

x2 (x 3)

(x 6)(x 1)(x 1)(x 9)

(x 9)(x 3)Next

Step #2: Divide the numerator and denominator by the common factors.

x2 (x 3)

(x 6)(x 1)(x 1)(x 9)

(x 9)(x 3)1

1

1

1

1

1

Step #3: Multiply the numerator and the denominator.

x2

x 6

Next: division of rational expressions.

Recall how to divide by a fraction:

Multiply by the reciprocal of the divisor.

4

5

16

25

4

5

25

16

4 25

516

1

1

5

4

5

4

AKA: Keep Flip Change

Ex: Simplifyy 2

y2 10 y 24

y2 2y

y2 2y 8

y 2

y2 10 y 24

y2 2y

y2 2y 8

y 2

y2 10 y 24y2 2y 8

y2 2y

y 2

(y 12)(y 2)(y 4)(y 2)

y(y 2)

1 1

1 1

Next

4

( 12)

y

y y

Keep - Flip - Change

Now you try to simplify the expression:

x 3

x2 4x 12

2x2 6x

x 2

Keep - Flip - Change

Answer: 1

2x(x 6)Now try these on your own.Keep - Flip - Change

1) x + 3

2x3 2x2

x2 7x 6

x2 10x 21

2) 3x 67x 7

5x 1014x 14

Here are the answers:

1) x 6

2x2 (x 7)

2) 6(x 1)5(x 1)