Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Section A.5
Rational Expressions
Math 1051 - Precalculus I
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
A.5 Rational Expressions
Factor: 6x 2 − 11x − 10
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
A.5 Rational Expressions
Factor: 6x 2 − 11x − 10
6x 2 − 11x − 10 = 6x 2 + 4x − 15x − 10
= (6x 2 + 4x) − (15x − 10)
= 2x(3x + 2) − 5(3x + 2)
= (3x + 2)(2x − 5)
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
Example of Not So Rational Expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
Example of Not So Rational Expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Example of Not So Rational Expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
Example of Not So Rational Expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
x3
The ratio of two polynomials, such as x 2 −2x+1
Example of Not So Rational Expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
x3
The ratio of two polynomials, such as x 2 −2x+1
Example of Not So Rational Expressions
Giant Shrimp.
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
x3
The ratio of two polynomials, such as x 2 −2x+1
Example of Not So Rational Expressions
Giant Shrimp.
The dog ate my homework.
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
x3
The ratio of two polynomials, such as x 2 −2x+1
Example of Not So Rational Expressions
Giant Shrimp.
The dog ate my homework.
I can do this stuff in my head.
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
x3
The ratio of two polynomials, such as x 2 −2x+1
Example of Not So Rational Expressions
Giant Shrimp.
The dog ate my homework.
I can do this stuff in my head.
Math is easy.
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational Expressionals
The more you study the more you learn.
A good night’s sleep helps you to think more clearly.
Space confines itself to a vast area.
x3
The ratio of two polynomials, such as x 2 −2x+1
Example of Not So Rational Expressions
Giant Shrimp.
The dog ate my homework.
I can do this stuff in my head.
Math is easy.
Expressions like
√
x
1
log(x)
|x|
2
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator and
denominator are both polynomials.
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator and
denominator are both polynomials. For example:
f (x) =
x4
x 3 − 2x 2 + x
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator and
denominator are both polynomials. For example:
f (x) =
x4
x 3 − 2x 2 + x
.
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator and
denominator are both polynomials. For example:
f (x) =
x4
x 3 − 2x 2 + x
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rational
expressions:
Reduce
Multiply
Divide
Add and Subtract
Simplify complex expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rational
expressions:
Reduce
Multiply
Divide
Add and Subtract
Simplify complex expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rational
expressions:
Reduce
Multiply
Divide
Add and Subtract
Simplify complex expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rational
expressions:
Reduce
Multiply
Divide
Add and Subtract
Simplify complex expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rational
expressions:
Reduce
Multiply
Divide
Add and Subtract
Simplify complex expressions
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Remember how we reduce fractions of numbers (rational
numbers):
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Remember how we reduce fractions of numbers (rational
numbers):
18
24
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Remember how we reduce fractions of numbers (rational
numbers):
18
24
204
238
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all the
polynomials, and then cancel like factors (but be careful about
the domain!):
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all the
polynomials, and then cancel like factors (but be careful about
the domain!):
9x
3x + 18
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all the
polynomials, and then cancel like factors (but be careful about
the domain!):
9x
3x + 18
x3 − 1
x3 − x2
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all the
polynomials, and then cancel like factors (but be careful about
the domain!):
9x
3x + 18
x3 − 1
x3 − x2
6x 3 − 2x 2 − 20x
4x 2 − 8x
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Remember how we multiply rational numbers:
a c
ac
· =
b d
bd
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Remember how we multiply rational numbers:
a c
ac
· =
b d
bd
5 6
·
4 7
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Again, we do the same thing for polynomials, but it’s easiest if
you leave everything in factored form:
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Again, we do the same thing for polynomials, but it’s easiest if
you leave everything in factored form:
2x + 8
3x 2
· 2
6x
x − 16
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
How do we multiply rational numbers?
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
How do we multiply rational numbers?
a
b
c
d
Rational Expressions
=
ad
bc
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
How do we multiply rational numbers?
a
b
c
d
=
ad
bc
4
3
2
11
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Again, we do the same thing (but we have to pay attention to
division by 0 problems):
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Again, we do the same thing (but we have to pay attention to
division by 0 problems):
x−1 x−3
x+1 · x+2
x 2 −3x
x 2 −1
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Adding and subtracting rational numbers:
a
c
ad + bc
+ =
b d
bd
c
ad − bc
a
− =
b d
bd
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Adding and subtracting rational numbers:
a
c
ad + bc
+ =
b d
bd
c
ad − bc
a
− =
b d
bd
But it might help to cancel things first, so we should look at the
factors
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Again, same thing with (but we have to pay attention to division
by 0 problems):
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Again, same thing with (but we have to pay attention to division
by 0 problems):
x2
2
x −1
− 2
−x −6 x −9
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Example with numbers:
2
3
3
5−1
3−
1+
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Example with numbers:
2
3
3
5−1
3−
1+
Example with polynomials:
2
x
1
x+1
3−
2+
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Moral
The moral of the story is:
Operations work just like they do with numbers
But sometimes it helps to factor first
Need to pay attention to divide by 0 issues
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Moral
The moral of the story is:
Operations work just like they do with numbers
But sometimes it helps to factor first
Need to pay attention to divide by 0 issues
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Moral
The moral of the story is:
Operations work just like they do with numbers
But sometimes it helps to factor first
Need to pay attention to divide by 0 issues
Rational Expressions
Section A.5
Reduce
Multiply
Divide
Add and Subtract
Simplifying Complex Expressions
Read section A.6 before next lecture.
Rational Expressions
Section A.5