View
2
Download
0
Category
Preview:
Citation preview
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Section A.5
Rational Expressions
Math 1051 - Precalculus I
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
A.5 Rational Expressions
Factor: 6x2 − 11x − 10
6x2 − 11x − 10 = 6x2 + 4x − 15x − 10= (6x2 + 4x)− (15x − 10)= 2x(3x + 2)− 5(3x + 2)= (3x + 2)(2x − 5)
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
A.5 Rational Expressions
Factor: 6x2 − 11x − 10
6x2 − 11x − 10 = 6x2 + 4x − 15x − 10= (6x2 + 4x)− (15x − 10)= 2x(3x + 2)− 5(3x + 2)= (3x + 2)(2x − 5)
Rational Expressions Section A.5
ReduceMultiply
DivideAdd 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.The ratio of two polynomials, such as x3
x2−2x+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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.
The ratio of two polynomials, such as x3
x2−2x+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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+1Example 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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+1Example 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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+1Example 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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+1Example 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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+1Example 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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Examples of Rational ExpressionalsThe 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.The ratio of two polynomials, such as x3
x2−2x+1Example 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
1|x |
log(x)√
x2
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator anddenominator are both polynomials.
For example:
f (x) =x4
x3 − 2x2 + x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator anddenominator are both polynomials. For example:
f (x) =x4
x3 − 2x2 + x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator anddenominator are both polynomials. For example:
f (x) =x4
x3 − 2x2 + x
.Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Rational Expressions
Rational Expressions are fractions where the numerator anddenominator are both polynomials. For example:
f (x) =x4
x3 − 2x2 + x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rationalexpressions:
ReduceMultiplyDivideAdd and SubtractSimplify complex expressions
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rationalexpressions:
ReduceMultiplyDivideAdd and SubtractSimplify complex expressions
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rationalexpressions:
ReduceMultiplyDivideAdd and SubtractSimplify complex expressions
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rationalexpressions:
ReduceMultiplyDivideAdd and SubtractSimplify complex expressions
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Operations with Rational Expressions
There are five basic operations we can do with rationalexpressions:
ReduceMultiplyDivideAdd and SubtractSimplify complex expressions
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Remember how we reduce fractions of numbers (rationalnumbers):
1824
204238
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Remember how we reduce fractions of numbers (rationalnumbers):
1824
204238
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Remember how we reduce fractions of numbers (rationalnumbers):
1824
204238
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all thepolynomials, and then cancel like factors (but be careful aboutthe domain!):
9x3x + 18
x3 − 1x3 − x2
6x3 − 2x2 − 20x4x2 − 8x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all thepolynomials, and then cancel like factors (but be careful aboutthe domain!):
9x3x + 18
x3 − 1x3 − x2
6x3 − 2x2 − 20x4x2 − 8x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all thepolynomials, and then cancel like factors (but be careful aboutthe domain!):
9x3x + 18
x3 − 1x3 − x2
6x3 − 2x2 − 20x4x2 − 8x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
We do the same thing for polynomials. First factor all thepolynomials, and then cancel like factors (but be careful aboutthe domain!):
9x3x + 18
x3 − 1x3 − x2
6x3 − 2x2 − 20x4x2 − 8x
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Remember how we multiply rational numbers:
ab· c
d=
acbd
54· 6
7
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Remember how we multiply rational numbers:
ab· c
d=
acbd
54· 6
7
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Again, we do the same thing for polynomials, but it’s easiest ifyou leave everything in factored form:
2x + 86x
· 3x2
x2 − 16
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Again, we do the same thing for polynomials, but it’s easiest ifyou leave everything in factored form:
2x + 86x
· 3x2
x2 − 16
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
How do we multiply rational numbers?
abcd=
adbc
432
11
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
How do we multiply rational numbers?
abcd=
adbc
432
11
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
How do we multiply rational numbers?
abcd=
adbc
432
11
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Again, we do the same thing (but we have to pay attention todivision by 0 problems):
x−1x+1 ·
x−3x+2
x2−3xx2−1
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Again, we do the same thing (but we have to pay attention todivision by 0 problems):
x−1x+1 ·
x−3x+2
x2−3xx2−1
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Adding and subtracting rational numbers:
ab+
cd
=ad + bc
bd
ab− c
d=
ad − bcbd
But it might help to cancel things first, so we should look at thefactors
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Adding and subtracting rational numbers:
ab+
cd
=ad + bc
bd
ab− c
d=
ad − bcbd
But it might help to cancel things first, so we should look at thefactors
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Again, same thing with (but we have to pay attention to divisionby 0 problems):
2x2 − x − 6
− x − 1x2 − 9
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Again, same thing with (but we have to pay attention to divisionby 0 problems):
2x2 − x − 6
− x − 1x2 − 9
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Example with numbers:
3− 23
1 + 35−1
Example with polynomials:
3− 2x
2 + 1x+1
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Example with numbers:
3− 23
1 + 35−1
Example with polynomials:
3− 2x
2 + 1x+1
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Moral
The moral of the story is:Operations work just like they do with numbersBut sometimes it helps to factor firstNeed to pay attention to divide by 0 issues
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Moral
The moral of the story is:Operations work just like they do with numbersBut sometimes it helps to factor firstNeed to pay attention to divide by 0 issues
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Moral
The moral of the story is:Operations work just like they do with numbersBut sometimes it helps to factor firstNeed to pay attention to divide by 0 issues
Rational Expressions Section A.5
ReduceMultiply
DivideAdd and Subtract
Simplifying Complex Expressions
Read section A.6 before next lecture.
Rational Expressions Section A.5
Recommended