Section A - University of Minnesotahankx003/Fall2012/Lectures/A5.pdf · The dog ate my homework. I...

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DivideAdd and Subtract

Simplifying Complex Expressions

Section A.5

Rational Expressions

Math 1051 - Precalculus I

Rational Expressions Section A.5

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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)


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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


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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



1|x |

log(x)√

x2


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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



1|x |

log(x)√

x2


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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



1|x |

log(x)√

x2


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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


1|x |

log(x)√

x2


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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


ReduceMultiply






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


ReduceMultiply






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


ReduceMultiply






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


ReduceMultiply







1|x |

log(x)√

x2


ReduceMultiply




Rational Expressions are fractions where the numerator anddenominator are both polynomials.

For example:

f (x) =x4

x3 − 2x2 + x


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Rational Expressions are fractions where the numerator anddenominator are both polynomials. For example:

f (x) =x4

x3 − 2x2 + x


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f (x) =x4

x3 − 2x2 + x

.Rational Expressions Section A.5

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f (x) =x4

x3 − 2x2 + x


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Operations with Rational Expressions

There are five basic operations we can do with rationalexpressions:

ReduceMultiplyDivideAdd and SubtractSimplify complex expressions


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Remember how we reduce fractions of numbers (rationalnumbers):

1824

204238


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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


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Remember how we multiply rational numbers:

ab· c

d=

acbd

54· 6

7


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Again, we do the same thing for polynomials, but it’s easiest ifyou leave everything in factored form:

2x + 86x

· 3x2

x2 − 16


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How do we multiply rational numbers?

abcd=

adbc

432

11


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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


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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


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Again, same thing with (but we have to pay attention to divisionby 0 problems):

2x2 − x − 6

− x − 1x2 − 9


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Example with numbers:

3− 23

1 + 35−1

Example with polynomials:

3− 2x

2 + 1x+1


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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


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Read section A.6 before next lecture.


Section A - University of Minnesotahankx003/Fall2012/Lectures/A5.pdf · The dog ate my homework. I...

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