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------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/master20.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADzgAAAAUAJQAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAEAAAALQEAAAQAAAAnAf//HAAAAPsC1P8A AAAAAACQAQAAAAAAQAAAQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQA AAAuARgABAAAAAIBAQAFAAAACQIAAAACJQAAADIKOQA2ARQAAABSYXRpb25hbCBFeHByZXNzaW9u cyAAGAAMAAsAGAAZABgACgANAB0AFgAZAA8AGAAXABYACgAZABgAFgAEAAAALgEAABwAAAD7AhAA BwAAAAAAvAIAAAAAAQICIlN5c3RlbQAAAAAAABgAAAAsnhMAAQAAAOQEAAAAAAAABAAAAC0BBAAE AAAA8AEDAAMAAAAAAA== ------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0171.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0172.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0173.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5, 
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0174.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5,  πx – 0.762 are polynomials.
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0175.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5,  πx – 0.762 are polynomials.
Rational (fractiona= l) expressions are expressions of the
form      , where P= and Q are polynomials.
P
Q
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0176.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5,  πx – 0.762 are polynomials.
Rational (fractiona= l) expressions are expressions of the
form      , where P= and Q are polynomials.
P
Q
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
All polynomials are rational expressions by viewing P as      .
P
1
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0177.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5,  πx – 0.762 are polynomials.
Rational (fractiona= l) expressions are expressions of the
form      , where P= and Q are polynomials.
P
Q
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
All polynomials are rational expressions by viewing P as      .
P
1
x – 2 = ;
x2 – 2 x + 1 ,
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0178.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5,  πx – 0.762 are polynomials.
Rational (fractiona= l) expressions are expressions of the
form      , where P= and Q are polynomials.
P
Q
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
All polynomials are rational expressions by viewing P as      .
P
1
x – 2 = ;
x2 – 2 x + 1 ,
x(x – 2) = ;
(x + 1) (2x + 1) = ;
 = are rational expressions.
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Rational Express= ions
For example,
7x12 – 4x5 + 6x – Ö5,  πx – 0.762 are polynomials.
Rational (fractiona= l) expressions are expressions of the
form      , where P= and Q are polynomials.
P
Q
Polynomials are expr= essions of the form
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:.37em;mso-text-r= aise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x+ an-2xn-2+…+= a1x1+ a0 
where the a’s are numbers.
All polynomials are rational expressions by viewing P as      .
P
1
x – 2 = ;
x2 – 2 x + 1 ,
x(x – 2) = ;
(x + 1) (2x + 1) = ;
 = are rational expressions.
x – 2 = ;
Ö<= /i> 2 x + 1
is not a = rational expression because the
denominat= or is not a polynomial.
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Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
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Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
is in the expanded form.
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0245.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) . = ;
is in the expanded form.
In the factored form, its
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0246.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) . = ;
x – 2 = ;
x2 + 1
The expre= ssion
is in bot= h forms.
is in the expanded form.
In the factored form, its
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0247.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) . = ;
x – 2 = ;
x2 + 1
The expre= ssion
is in bot= h forms.
is in the expanded form.
In the factored form, its
Example A.<= span style=3D'mso-spacerun:yes'>  Put the following expressions in = the factored form.
a.
x2 – 3x – 10
x2 – 3x
=3D<= span lang=3DEN-US style=3D'mso-hansi-font-family:Arial;font-size:133%;color:#00= 3300; mso-fareast-language:ZH-CN;mso-special-format:lastCR;display:none'>
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Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) . = ;
x – 2 = ;
x2 + 1
The expre= ssion
is in bot= h forms.
is in the expanded form.
In the factored form, its
Example A.<= span style=3D'mso-spacerun:yes'>  Put the following expressions in = the factored form.
(x – 5)(x + 2)
a.
  &= nbsp; x(x – 3)
x2 – 3x – 10
x2 – 3x
=3D<= span lang=3DEN-US style=3D'mso-hansi-font-family:Arial;font-size:133%;color:#00= 3300; mso-fareast-language:ZH-CN;mso-special-format:lastCR;display:none'>
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Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) . = ;
x – 2 = ;
x2 + 1
The expre= ssion
is in bot= h forms.
is in the expanded form.
In the factored form, its
Example A.<= span style=3D'mso-spacerun:yes'>  Put the following expressions in = the factored form.
(x – 5)(x + 2)
a.
  &= nbsp; x(x – 3)
x2 – 3x – 10
x2 – 3x
=3D<= span lang=3DEN-US style=3D'mso-hansi-font-family:Arial;font-size:133%;color:#00= 3300; mso-fareast-language:ZH-CN;mso-special-format:lastCR;display:none'>
b.
x2 – 3x + 10
x2 – 3
is in the factored form
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Rational Express= ions
Just as polynomials ma= y be given in the expanded form or factored form, rational expressions also be given in these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) . = ;
x – 2 = ;
x2 + 1
The expre= ssion
is in bot= h forms.
is in the expanded form.
In the factored form, its
Example A.<= span style=3D'mso-spacerun:yes'>  Put the following expressions in = the factored form.
(x – 5)(x + 2)
a.
  &= nbsp; x(x – 3)
x2 – 3x – 10
x2 – 3x
=3D<= span lang=3DEN-US style=3D'mso-hansi-font-family:Arial;font-size:133%;color:#00= 3300; mso-fareast-language:ZH-CN;mso-special-format:lastCR;display:none'>
b.
x2 – 3x + 10
x2 – 3
is in the factored form
Operations for rational expressions are similar to operations for fract= ions.
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Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
------=_NextPart_01C970E2.6A815370 Content-Location: file:///C:/2D664A33/4OperationsofRationalExpressions_files/slide0253.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Rational Expressions
PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:133%;mso-fareast-language:ZH-CN'> then cancel the common factors.&#= 13;
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
Example B.<= span style=3D'mso-spacerun:yes'>  Reduce the following expressions.=
(x – 2)(x + 3)
(x + 3)(x + 2)
a.
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
Example B.<= span style=3D'mso-spacerun:yes'>  Reduce the following expressions.=
(x – 2)(x + 3)
It's factor= ed, proceed to cancel the common factor.
(x + 3)(x + 2)
a.
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
Example B.<= span style=3D'mso-spacerun:yes'>  Reduce the following expressions.=
(x – 2)(x + 3)
It's factor= ed, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
a.
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
Example B.<= span style=3D'mso-spacerun:yes'>  Reduce the following expressions.=
(x – 2)(x + 3)
1
=3D
x – 2&n= bsp;
x + 2
It's factor= ed, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
Example B.<= span style=3D'mso-spacerun:yes'>  Reduce the following expressions.=
(x – 2)(x + 3)
1
=3D
x – 2&n= bsp;
x + 2
It's factor= ed, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
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PN
QN
=3D
=3D
1
Rational Express= ions
Cancellat= ion Law:  Common factor may be canceled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if it is canceled as much as possible. 
To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
Example B.<= span style=3D'mso-spacerun:yes'>  Reduce the following expressions.=
(x – 2)(x + 3)
1
=3D
x – 2&n= bsp;
x + 2
It's factor= ed, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
It's in the facto= red form. There is no common factor.
So its already reduced
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Rational Express= ions
c.
x2 – 1
x2 – 3x+ 2
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Rational Express= ions
c.
x2 – 1
x2 – 3x+ 2
Factor then cancel
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Rational Express= ions
c.
x2 – 1
x2 – 3x+ 2