MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C8196E.45924550" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C8196E.45924550 Content-Location: file:///C:/F1764A01/SyntheticDivisionSyntheticSubstitutionA.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Synthetic Division or Synthetic Substitution for Polynomial Function= s

 

Synthetic Division or Synthetic Substitu= tion for Polynomial Functions

 

********************************************

 

Definition:  = An artificial (synthetic) procedure = used in place of Long Division or Algebraic Substitution.

 

Example Polynomial Function:          F(X) =3D 2= X3 +3X2 –11X –6

 

Synthetic Division is an artificial procedure to locate = points in an Infinite Solution Set of a Polynomial.

It also is a means to determine special <= b>points called zero or roots or intercept points in a Solution Set.

The procedure uses detached coefficien= ts from the polynomial and arranges them in a horizontal row

with an X value to be synthetic divided into the= coefficients.  The steps to this procedure and co= ncrete

examples of using the procedure are provided it = the paragraph and examples that follow below.

 

After the horizontal arrangement has been placed= to the right of any X value the first step is to place

a 0 under the first coefficient then add.=   The next step is to the thi= s sum and multiply it by the X value

and place this product under the secon= d coefficient then add.  = Again this sum is multiplied by the X

value and the second product is placed un= der the third coefficient.  This procedure is repeated until

all coefficients have been added.  If the last number is any <= b>nonzero number then it is a normal point

in the Infinite Solution Set of the Polynomial Function with a Y value equal to the final number of SD.<= /p>

If the last number is a zero then it is a special point in the SS called a root or intercept point of the SS.

 

 

  &nbs= p;             = [ +1]      &nbs= p; +2        &= nbsp;   +3        &= nbsp;   –11        &= nbsp; –6        &= nbsp;           &nbs= p;       Coefficients of Original Polynomial Function

  &nbs= p;            &= nbsp;           &nbs= p;     0&n= bsp;            +2        &= nbsp;   +5        &= nbsp;   –6

  &nbs= p;            &= nbsp;           &nbs= p;    +2        &= nbsp;   +5        &= nbsp;   –6        &= nbsp;   –12        &= nbsp;           &nbs= p;     An X value of +1 yields a Y value of –12

  &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;          &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;  thus a normal point in SS is (+1, –12)

  &nbs= p;             = [+2]         = +2        &= nbsp;   +3        &= nbsp;   –11        &= nbsp; –6

  &nbs= p;            &= nbsp;           &nbs= p;     0&n= bsp;            +4        &= nbsp;   +14        &= nbsp; +6

  &nbs= p;            &= nbsp;           &nbs= p;    +2        &= nbsp;   +7        &= nbsp;   +3        &= nbsp;   0       =             &nb= sp;         An X value of +2 yields a Y value of 0

  &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;            thus a special point in SS is (+2, 0)

 

 

Since the second Synthetic Division yields a 0 t= hen the numbers in front of it are coefficients of a

Depressed Equation or would be the Quotient of a normal long Algebraic Division.  To determine

the final two zero numbers, repeat the Synthetic Division procedure on the Depressed Equation or if

the coefficients are only (3) then the Depressed Equation then it can be factored or the QRF is used.

 

 

  &nbs= p;             = [–1]         = +2        &= nbsp;   +7        &= nbsp;   +3        &= nbsp;           &nbs= p;            &= nbsp;          [–= 2]         = +2        &= nbsp;   +7        &= nbsp;   +3

  &nbs= p;            &= nbsp;           &nbs= p;     0&n= bsp;            –2        &= nbsp;   –10        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            0       =       –4        &= nbsp;   –6

  &nbs= p;            &= nbsp;           &nbs= p;    +2        &= nbsp;   +5        &= nbsp;   –7  No good           &nbs= p;            &= nbsp;           &nbs= p;     +2        &= nbsp;   +3        &= nbsp;   –3  No good

 

  &nbs= p;             = [+3]         = +2        &= nbsp;   +7        &= nbsp;   +3        &= nbsp;           &nbs= p;            &= nbsp;          [–= 3]         = +2        &= nbsp;   +7        &= nbsp;   +3

  &nbs= p;            &= nbsp;           &nbs= p;     0&n= bsp;            +6        &= nbsp;   +26        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            0       =       –6        &= nbsp;   –3

  &nbs= p;            &= nbsp;           &nbs= p;    +2        &= nbsp;   +13        &= nbsp; +29 No Good      =             &nb= sp;            =           +2        &= nbsp;   +1        &= nbsp;   0 

Therefore  (–3,0) and (–1/2,0)

  &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =           

Now, the process of finding all the intercepts or roots or zero numbers might have been easier if

factoring had been used.   +2x2 +7x +3 =3D (X+3)(2x+1)<= span style=3D'mso-spacerun:yes'>  or X=3D–3 and X=3D –1/= 2   Thus the special<= /span>

or roots or intercepts of the Polynomial Function are:  (X,Y) =3D (+2,0)  , (–3,0) and (–1/2,0).=   Using

Synthetic Division to find a few more normal poi= nts provides enough data to sketch the Solution Set.

 

 

Tom Love=         =             &nb= sp;       Malone College         =             &nb= sp;    Fall 2007

 

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