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Methods of Divis= ion
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Methods of Divis= ion
The quest= ion of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of= this chapter. 
------=_NextPart_01C84742.9B085F90 Content-Location: file:///C:/32CB324E/MethodsofDivision_files/slide0354.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Methods of Divis= ion
We start with two division algorithms for polynomials of a single variable x; the long division<= /span> and synthetic = division.
The quest= ion of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of= this chapter. 
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Methods of Divis= ion
We start with two division algorithms for polynomials of a single variable x; the long division<= /span> and synthetic = division.
The quest= ion of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of= this chapter. 
The long division of polynomials is analogous to= <= span lang=3DEN-US style=3D'mso-bidi-font-family:Arial;mso-hansi-font-family:Ari= al; font-size:156%;mso-fareast-language:ZH-CN'>the long division of numbers and it’s a general division method for P(x) ÷ D(x).
------=_NextPart_01C84742.9B085F90 Content-Location: file:///C:/32CB324E/MethodsofDivision_files/slide0355.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Methods of Divis= ion
We start with two division algorithms for polynomials of a single variable x; the long division<= /span> and synthetic = division.
The quest= ion of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of= this chapter. 
The long division of polynomials is analogous to= <= span lang=3DEN-US style=3D'mso-bidi-font-family:Arial;mso-hansi-font-family:Ari= al; font-size:156%;mso-fareast-language:ZH-CN'>the long division of numbers and it’s a general division method for P(x) ÷ D(x).
The synthetic division is a simpler method for dividing polynomials P(x) by monomials of the form = ;
(x – c), i.e. P(x) ÷ (x – c= ).
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Methods of Divis= ion
We start with two division algorithms for polynomials of a single variable x; the long division<= /span> and synthetic = division.
The quest= ion of factorability and finding roots of real polynomials (polynomials with real coefficients) are the main themes of= this chapter. 
The long division of polynomials is analogous to= <= span lang=3DEN-US style=3D'mso-bidi-font-family:Arial;mso-hansi-font-family:Ari= al; font-size:156%;mso-fareast-language:ZH-CN'>the long division of numbers and it’s a general division method for P(x) ÷ D(x).
The synthetic division is a simpler method for dividing polynomials P(x) by monomials of the form = ;
(x – c), i.e. P(x) ÷ (x – c= ). This is particularly useful for checking for possible roots or finding = remainders of = the division. 
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The Long Division
 D(x)
 P(x)
Dividend
Divisor
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The Long Division
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
 D(x)
 P(x)
Dividend
Divisor
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 D(x)
 P(x)
Dividend
Divisor
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 D(x)
 P(x)
Dividend
Divisor
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 D(x)
 P(x)
Dividend
Divisor
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
 2x
 D(x)
 P(x)
Dividend
Divisor
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
 D(x)
 P(x)
Dividend
Divisor
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
– )
 D(x)
 P(x)
Dividend
Divisor
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
– )
– 
+ 
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
– )
– 
+ 
5x + 4
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
new <= /span>
dividend
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
4. Use this difference from the subtraction as the new dividend , repeat step 2 and 3. 
– )
– 
+ 
5x + 4
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
new <= /span>
dividend
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
4. Use this difference from the subtraction as the new dividend , repeat step 2 and 3. 
– )
– 
+ 
5x + 4
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
new <= /span>
dividend
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
4. Use this difference from the subtraction as the new dividend , repeat step 2 and 3. 
– )
– 
+ 
5x + 4
5x – 20
– )
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
new <= /span>
dividend
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
4. Use this difference from the subtraction as the new dividend , repeat step 2 and 3. 
– )
– 
+ 
5x + 4
5x – 20
– )
– 
+ 
24
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
new <= /span>
dividend
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
Example A: Divide        = ;            &n= bsp;   using long division.  &n= bsp;            = ;       
 x – 4
 2x2 – 3x + 4
1. Set up the long division        = ;            &n= bsp;
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
3. Multiply this quotient to the divisor and subtract the result from the dividend.        = ;            &n= bsp; 
4. Use this difference from the subtraction as the new dividend , repeat step 2 and 3. 
5. Stop when the degree of the new dividend is smaller than the degree of the divisor.  <= /div>
– )
– 
+ 
5x + 4
5x – 20
– )
– 
+ 
24
Set up for the division P= (x) ÷ D(x) the same way as the dividing numbers. = ;
 D(x)
 P(x)
Dividend
Divisor
new <= /span>
dividend
2. Enter the quotient of the leading terms on the top.        = ;            &n= bsp;
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The Long Division
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
5x + 4
5x – 20
24
Here are the name= s of the terms:
dividend P(x)
divisor D(x)
quotient Q(x)
remainder R(x)
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The Long Division
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
5x + 4
5x – 20
24
Here are the name= s of the terms:
dividend P(x)
divisor D(x)
quotient Q(x)
remainder R(x)
We check easily t= hat
 x – 4
 2x2 – 3x + 4
 =3D
 2x + 5
+
 x – 4   
24
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The Long Division
 2x2 – 3x + 4
 x – 4
 2x + 5
 2x2 – 8x
5x + 4
5x – 20
24
Here are the name= s of the terms:
dividend P(x)
divisor D(x)
quotient Q(x)
remainder R(x)
We check easily t= hat
 x – 4
 2x2 – 3x + 4
 =3D
 2x + 5
+
 x – 4   
24
We summarize the end resu= lt from performing the long division algorithm in the following theorem.
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The Long Division
Long Division Theorem:  Using long division for
P(x) ÷ D(x), we may obtain a quotient <= /span>Q(x) and a = reminder R(x) such that
------=_NextPart_01C84742.9B085F90 Content-Location: file:///C:/32CB324E/MethodsofDivision_files/slide0376.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
The Long Division
Long Division Theorem:  Using long division for
P(x) ÷ D(x), we may obtain a quotient <= /span>Q(x) and a = reminder R(x) such that
 D(x)
 P(x)
 =3D Q(x) +
 D(x)
 R(x)
with deg R(x) < deg D(x).
------=_NextPart_01C84742.9B085F90 Content-Location: file:///C:/32CB324E/MethodsofDivision_files/slide0376_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C84742.9B085F90 Content-Location: file:///C:/32CB324E/MethodsofDivision_files/slide0376_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C84742.9B085F90 Content-Location: file:///C:/32CB324E/MethodsofDivision_files/slide0377.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
The Long Division
Long Division Theorem:  Using long division for
P(x) ÷ D(x), we may obtain a quotient <= /span>Q(x) and a = reminder R(x) such that
 D(x)
 P(x)
 =3D Q(x) +
 D(x)
 R(x)
with deg R(x) < deg D(x).
Example B: Divide=
 x2 + 1
 x3 – 2x2 + 3
and write it in the =
form of        = ;            
Q(x) +
 D(x)
 R(x)
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The Long Division
Long Division Theorem:  Using long division for
P(x) ÷ D(x), we may obtain a quotient <= /span>Q(x) and a = reminder R(x) such that
 D(x)
 P(x)
 =3D Q(x) +
 D(x)
 R(x)
with deg R(x) < deg D(x).
Example B: Divide=
 x2 + 1
 x3 – 2x2 + 3
and write it in the =
form of        = ;            
 x3 – 2x2  + 0x + 3<= /div>
 x2 + 1
Q(x) +
 D(x)
 R(x)
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The Long Division
Long Division Theorem:  Using long division for
P(x) ÷ D(x), we may obtain a quotient <= /span>Q(x) and a = reminder R(x) such that
 D(x)
 P(x)
 =3D Q(x) +
 D(x)
 R(x)