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