MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C8196E.1B5AC690" 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.1B5AC690 Content-Location: file:///C:/0E3A5172/PolynomialFunctionsUnitThreeARRIR.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Creating Polynomial Functions (Equations) from Rational Roots - Unit= (2)

Creating Polynomial Functions (Equations) from Rational & Irrational Roots

 

***********= *********************************

 

Given Rational and Irrational Roots:        &= nbsp;  ( + 2 )  ,  ( 2 +   )  ,=   ( 2 –   )

 

<= ![endif]>        &= nbsp;           &nbs= p;    

 

        &= nbsp;                  -4       -3      -2     -1       0 +1    +2       +3   +4

 

Roots:    X =3D 2 –     &= nbsp;           &nbs= p;             =      X =3D + 2        =      X =3D 2 +  

 

Factors:  ( X ­– 2 )        &= nbsp;           &nbs= p;          X2  –  ( R1 + R2 ) = X + ( R1R2)  =3D  X2 ­= ;– 4X ­– 1

 

Equat= ion (Function):    Y =3D = + X3 – 5X2 + 7X + 2

&nb= sp;

Changing all signs generates an equivalent = but negative polynomial:  Y =3D ­–  X3 + 5X2 &sh= y; 7X ­ 2

&nb= sp;

To do th= is (­–) equivalent generation change all signs of (x) terms  then  to check factor the equation.

 

<= o:p> 

***********= *********************************

 

Given Rational and Irrational Roots:        &= nbsp;  ( + 2 )  ,  ( 3 +   )  ,=   ( 3 –   )

 

<= ![endif]>        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           

 

        &= nbsp;             =       -4     -= 3      -2      -1       0      +1     +2     +3     +4

 

Roots:       X =3D=   – 2           X = =3D  3 –            = X =3D 3 +     

 

Factors:  ( X + 2 )           &nb= sp;            =        X2  –  ( R1 + R2 ) X + ( R1R2)  =3D  X2 ­+ 6X ­+ 7

 

Equat= ion (Function):   Y =3D= + X3 + 5X2 + X – 7

 

Changing all signs generates an equivalent = but negative polynomial:  Y =3D – X3 – 5X2 – X + 7

&nb= sp;

To do th= is (­–) equivalent generation change all signs of (x) terms  then  to check factor the equation.

 

***********= *********************************

 

Creating Polynomial Functi= ons with Ration and Irrational Roots tends to be a little more difficult

but using the handy dandy = Abstract Trinomial generator  X2  –  ( R1 + R2 ) X + ( R1R2)=   does help!

 

It is derived from (X – R1 ) ( X – R2 ) =3D ( X2 – R1X  = – R2 X + R1 R2 ) =3D = X2  – = ( R1 + R2 ) = X + R1R2.

Since Given Roots of:   = R1 & R2    thus  X =3D + R1  and   X =3D + R2   and therefore  (= X – R1 ) ( X – R2 ).

 

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

=  

 

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Tom Love=         =             &nb= sp;       Malone College        =             &nb= sp;      Fall 2003

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