rational expressions & equations. what is a rational expression it is a monomial which is a...

Rational Expressions & Equations

What is a Rational Expression

It is a Monomial which is a number or letter(variable) or combination.

3x15a2b

8a2b2x5/3y

u34

€

x 2 + 3x −10

3x + 2OR IT’S A POLYNOMIAL QUOTIENT

€

3⋅ xx

= ? vs 3 + x

x= ?

Reduce With Caution

€

3⋅ xx

= 3 vs 3+ x

x=

3

x+x

x=

3

x+1

3⋅33

= 3 vs 3 + 3

3=

6

3= 2

Reduce With Caution

€

6m −14

7m+m

7m=

7m −14

7m≠ −14

Reduce With Caution

€

6m −14

7m+m

7m=

7m −14

7m≠ −14

7m −14

7m=

7(m − 2)

7m=m − 2

m

Think of a Factor Tree

96

4 12

6222

32222

What is a Simplifying

€

9

6=

3⋅33⋅2

What is a Simplifying

€

9

6=

3⋅33⋅2

=3

2

What is a Simplifying

€

3x 2

6x=

3⋅ x ⋅ x3⋅2 ⋅ x

=x

2

Simplifying Rational Expressions

FACTOR FIRST€

5x 2 −10x

5x=

5x(x − 2)

5x

Simplifying Rational Expressions

The Key is to:

FACTOR & REDUCE€

5x 2 −10x

5x=

5x(x − 2)

5x= x − 2

FACTOR

€

5x −10

5x=

REDUCE

€

5x −10

5x=

5(x − 2)

5x=x − 2

x

Normally x can be any real # but since you can’tdivide by 0 x can be any real # except 0. For x=0 this algebraic expression is UNDEFINED.

FACTOR

€

5x 2 −10x

5x=

5x(x − 2)

5x

REDUCE

€

5x 2 −10x

5x=

5x(x − 2)

5x= x − 2

Simplify

€

y 2 + 3y + 2

y 2 −1

Simplifying

Polynomials

Simplify

€

y 2 + 3y + 2

y 2 −1=

(y + 2)(y +1)

(y +1)(y −1)

2

3

21

€

y 2 + 3y + 2

y 2 −1

Simplifying

Polynomials

1. Factor numerator & denom.

Simplify

€

y 2 + 3y + 2

y 2 −1=

(y + 2)(y +1)

(y +1)(y −1)

2

3

21

€

y 2 + 3y + 2

y 2 −1

Simplifying

Polynomials

1. Factor numerator & denom.

2. Reduce

€

( y + 2 )( y + 1 )

( y + 1 )( y − 1 )

=

( y + 2 )

( y − 1 )

Simplify

€

y 2 + 3y + 2

y 2 −1=

(y + 2)(y +1)

(y +1)(y −1)

2

3

21

€

y 2 + 3y + 2

y 2 −1

Simplifying

Polynomials

€

( y + 2 )( y + 1 )

( y + 1 )( y − 1 )

=

( y + 2 )

( y − 1 )

For y=1 this algebraic expression is UNDEFINED.

1. Factor numerator & denom.

2. Reduce

3. Name undefined values(Where denominator is 0)

FACTOR FIRST

€

y 2 + 3y + 2

y 2 −1=

(y + 2)(y +1)

(y +1)(y −1)

2

3

21

NOW REDUCE

€

(y + 2)(y +1)

(y +1)(y −1)=

(y + 2)

(y −1)For y=1 this algebraic expression is UNDEFINED.

Now Try These

€

2y 2 + 24y

8y=

€

2x 2 + 4x

8x 2 + 6x=

€

x 2 −1

2x 2 − x −1=

FACTOR & REDUCE

Answers

€

2y 2 + 24y

8y=

2y(y +12)

2 ⋅4 ⋅ y=y +12

4

€

2x 2 + 4x

8x 2 + 6x=

2x(x + 2)

2x(4x + 3)=

(x + 2)

(4x + 3)

€

x 2 −1

2x 2 − x −1=

(x +1)(x −1)

(2x +1)(x −1)=

(x +1)

(2x +1)

Are there any values which are UNDEFINED?

Undefined

€

2x 2 + 4x

8x 2 + 6x=

2x(x + 2)

2x(4x + 3)=

(x + 2)

(4x + 3)

€

x 2 −1

2x 2 − x −1=

(x +1)(x −1)

(2x +1)(x −1)=

(x +1)

(2x +1)

When is 2x+1=0? When x=-1/2

When is 4x+3=0? When x=-3/4

Can You Reduce This?

€

x − 4

4 − x=

Factor Out a Neg. 1

€

x − 4

4 − x=

−1(−x + 4)

4 − x=

−1(4 − x)

4 − x= −1

€

x − 4

4 − x= −1

New Try These

€

y − 6

6 − y=

€

9 − 6

6 − 9=

€

28 −1

1− 28=

€

8(4y − 2)

−2(2 − 4y)=