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Sign Charts and Inequalities II
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0244.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0245.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
N1
N2
Nk
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0246.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x).
N1
N2
Nk
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0247.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely.
N1
N2
Nk
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0248.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0249.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example A: P(x) =3D 2x – 16x5 + 32x3 
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0250.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example = A: P(x) =3D 2x – 16x5 + 32x3 
        = ;            &n= bsp;      =3D 2x(x4 – 8x2 + 16)
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0251.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example = A: P(x) =3D 2x – 16x5 + 32x3 
        = ;            &n= bsp;      =3D 2x(x4 – 8x2 + 16) =3D 2x3(x2 – 4)2
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0252.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example = A: P(x) =3D 2x – 16x5 + 32x3 
        = ;            &n= bsp;      =3D 2x(x4 – 8x2 + 16) =3D 2x3(x2 – 4)2
        = ;            &n= bsp;      =3D 2x(x + 2)2(x – 2)
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0253.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order of the ro= ots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example = A: P(x) =3D 2x – 16x5 + 32x3 
        = ;            &n= bsp;      =3D 2x(x4 – 8x2 + 16) =3D 2x3(x2 – 4)2
        = ;            &n= bsp;      =3D 2x(x + 2)2(x – 2)
Hence P(= x) is factorable with roots x =3D 0, 2, and –2.
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0254.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example = A: P(x) =3D 2x – 16x5 + 32x3 
        = ;            &n= bsp;      =3D 2x(x4 – 8x2 + 16) =3D 2x3(x2 – 4)2
        = ;            &n= bsp;      =3D 2x3(x + 2)2(x – 2)
Hence P(= x) is factorable with roots x =3D 0, 2, and –2.
x =3D 0 = has order 3,
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0255.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
In this section we give a theorem about sign-charts of factor= able polynomials and rational expressions.
Sign Charts and Inequalities II
Let P(x) =3D a<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>nxn + an-1x + .. + a1x + a0 be a polynomial that may be factored as
P(x) =3D an(x – r1)  (x – r2) .. (x &= #8211; rk) . 
Hence r1, r2,.. ,rk as the roots of P(x). The exponent= s = N<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:.37em;mso-text-= raise: -25%;mso-fareast-language:ZH-CN'>1, N2, .., Nk are called the order= of the r= oots respectiv= ely. Such polynomial are called factorable.
N1
N2
Nk
Example = A: P(x) =3D 2x – 16x5 + 32x3 
        = ;            &n= bsp;      =3D 2x(x4 – 8x2 + 16) =3D 2x3(x2 – 4)2
        = ;            &n= bsp;      =3D 2x3(x + 2)2(x – 2)
Hence P(= x) is factorable with roots x =3D 0, 2, and –2.
x =3D 0 = has order 3, x =3D 2 and –2 each has order <= span lang=3DEN-US style=3D'font-size:156%;color:#0000CC;mso-fareast-language:ZH= -CN'>2.
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Sign Charts and Inequalities II
An important property of a root is whether it is an = even-orde= red root or an odd-ordered root.
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Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
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Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
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Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
b. x =3D 2 or –2= are even-ordered roots
(each has order 2).
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0259.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
b. x =3D 2 or –2= are even-ordered roots
(each has order 2).
Theorem<= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>: For sig= n-charts of factorable polynomials
1. The signs are the s= ame on two sides of
even-ordered roots.
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Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
b. x =3D 2 or –2= are even-ordered roots
(each has order 2).
Theorem<= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>: For sig= n-charts of factorable polynomials
1. The signs are the s= ame on two sides of
even-ordered roots. (T= he sign stays the same across odd-ordered roots.)
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Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
b. x =3D 2 or –2= are even-ordered roots
(each has order 2).
Theorem<= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>: For sig= n-charts of factorable polynomials
1. The signs are the s= ame on two sides of
even-ordered roots. (T= he sign stays the same across odd-ordered roots.) &#= 13;
2. The signs are diffe= rent on two sides of
odd-ordered roots.
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0261.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
b. x =3D 2 or –2= are even-ordered roots
(each has order 2).
Theorem<= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>: For sig= n-charts of factorable polynomials
1. The signs are the s= ame on two sides of
even-ordered roots. (T= he sign stays the same across odd-ordered roots.) &#= 13;
2. The signs are diffe= rent on two sides of
odd-ordered roots. (Th= e sign switches across odd ordered root.)
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Sign Charts and Inequalities II
An important property = of a root is whether it is an even-ordered root or an odd-ordered root. 
For example, for P(x) = =3D 2x3(x + 2)2(x – 2)2,
a. x =3D 0 is an odd-o= rdered root (it's order is 3)
b. x =3D 2 or –2= are even-ordered roots
(each has order 2).
Theorem<= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>: For sig= n-charts of factorable polynomials
1. The signs are the s= ame on two sides of
even-ordered roots. (T= he sign stays the same across odd-ordered roots.) &#= 13;
2. The signs are diffe= rent on two sides of
odd-ordered roots. (Th= e sign switches across odd ordered root.)
This theo= rem simplifies the construction of sign-chart.
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Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
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Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  =
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Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0266.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0266_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wELAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AQoAgQAAAAAAALvg4wECAwK4hI+py+3Pggyw2ouz3vxM2oXiSJbRZKbqykKBAAtgS9d288bzzfd1 Dtv5hkQWUFdMKi3H4PIJjcSQUZOwGmpSsVyfVtYND79X8QVl+KjX7Lb7DY9Pp/G6/Y7P6/f7uY4P GCg4SPjmF1SYmAig2Hh3KOMoOUmJB1mJman5cbnp+WGG8QUWWppCZppqdal6htb6oFUGSxvrN1ub i3Or6yDR60sHPCz1R3yscISLzOzx2kwMCt0sPa1QAAA7 ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0267.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive.
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0267_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wELAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AQoAgQAAAAAAALvg4wECAwK4hI+py+3Pggyw2ouz3vxM2oXiSJbRZKbqykKBAAtgS9d288bzzfd1 Dtv5hkQWUFdMKi3H4PIJjcSQUZOwGmpSsVyfVtYND79X8QVl+KjX7Lb7DY9Pp/G6/Y7P6/f7uY4P GCg4SPjmF1SYmAig2Hh3KOMoOUmJB1mJman5cbnp+WGG8QUWWppCZppqdal6htb6oFUGSxvrN1ub i3Or6yDR60sHPCz1R3yscISLzOzx2kwMCt0sPa1QAAA7 ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0275.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>Mark the segment as such.  =
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0275_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wELAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AQoAgQAAAAAAALvg4wECAwK4hI+py+3Pggyw2ouz3vxM2oXiSJbRZKbqykKBAAtgS9d288bzzfd1 Dtv5hkQWUFdMKi3H4PIJjcSQUZOwGmpSsVyfVtYND79X8QVl+KjX7Lb7DY9Pp/G6/Y7P6/f7uY4P GCg4SPjmF1SYmAig2Hh3KOMoOUmJB1mJman5cbnp+WGG8QUWWppCZppqdal6htb6oFUGSxvrN1ub i3Or6yDR60sHPCz1R3yscISLzOzx2kwMCt0sPa1QAAA7 ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0275_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0268.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. = Mark the segment as such.  By the theorem, across the root x =3D 3 the sign change= s to "–" because x =3D 3 is odd-ordered.  <= /div>
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0268_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wELAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AQoAgQAAAAAAALvg4wECAwK4hI+py+3Pggyw2ouz3vxM2oXiSJbRZKbqykKBAAtgS9d288bzzfd1 Dtv5hkQWUFdMKi3H4PIJjcSQUZOwGmpSsVyfVtYND79X8QVl+KjX7Lb7DY9Pp/G6/Y7P6/f7uY4P GCg4SPjmF1SYmAig2Hh3KOMoOUmJB1mJman5cbnp+WGG8QUWWppCZppqdal6htb6oFUGSxvrN1ub i3Or6yDR60sHPCz1R3yscISLzOzx2kwMCt0sPa1QAAA7 ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0268_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0274.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. = Mark the segment as such.  By the theorem, across the root x =3D 3 the sign change= s to "–" because x =3D 3 is odd-ordered.  <= /div>
change
sign
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0274_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wETAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AREAgQAAAAAAALvg4/8AAALYhI+py+0Po5y02ouz3nH4D4aix5XmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9xn4BvoOsFhsRdstgYEbEE5fVy333CknE2vN+9zPZGP 52cE6IaVN0UYKAiUeLjY03iFZkBWaXmJmam5ydnm6cYZKjpKWmp6avo5h8ra6voKq6mKF1tbC2Cb OzoLquv7C7w7G0xcbEzGe6xciZj8+BP5DO2sNklF6Ch9g63NqJoN9ZXFB95dQ15ufu6Zrg5p7a5D Fg8Nz1UAADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0274_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0269.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. = Mark the segment as such.  By the theorem, across the root x =3D 3 the sign change= s to "–" because x =3D 3 is odd-ordered.  Similarly, acro= ss the root x =3D 0, the sign changes again to "+". 
change
sign
change
sign
+
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0269_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wETAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AREAgQAAAAAAALvg4/8AAALYhI+py+0Po5y02ouz3nH4D4aix5XmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9xn4BvoOsFhsRdstgYEbEE5fVy333CknE2vN+9zPZGP 52cE6IaVN0UYKAiUeLjY03iFZkBWaXmJmam5ydnm6cYZKjpKWmp6avo5h8ra6voKq6mKF1tbC2Cb OzoLquv7C7w7G0xcbEzGe6xciZj8+BP5DO2sNklF6Ch9g63NqJoN9ZXFB95dQ15ufu6Zrg5p7a5D Fg8Nz1UAADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0269_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0270.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. = Mark the segment as such.  By the theorem, across the root x =3D 3 the sign change= s to "–" because x =3D 3 is odd-ordered.  Similarly, acro= ss the root x =3D 0, the sign changes again to "+".  But across x = =3D -3 the sign stays as "+" because it even-ordered
change
sign
change
sign
+
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0270_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wETAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AREAgQAAAAAAALvg4/8AAALYhI+py+0Po5y02ouz3nH4D4aix5XmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9xn4BvoOsFhsRdstgYEbEE5fVy333CknE2vN+9zPZGP 52cE6IaVN0UYKAiUeLjY03iFZkBWaXmJmam5ydnm6cYZKjpKWmp6avo5h8ra6voKq6mKF1tbC2Cb OzoLquv7C7w7G0xcbEzGe6xciZj8+BP5DO2sNklF6Ch9g63NqJoN9ZXFB95dQ15ufu6Zrg5p7a5D Fg8Nz1UAADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0270_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0271.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. = Mark the segment as such.  By the theorem, across the root x =3D 3 the sign change= s to "–" because x =3D 3 is odd-ordered.  Similarly, acro= ss the root x =3D 0, the sign changes again to "+".  But across x = =3D -3 the sign stays as "+" because it even-ordered
change
sign
change
sign
+
sign
unchanged
+
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0271_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wETAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AREAgQAAAAAAALvg4/8AAALYhI+py+0Po5y02ouz3nH4D4aix5XmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9xn4BvoOsFhsRdstgYEbEE5fVy333CknE2vN+9zPZGP 52cE6IaVN0UYKAiUeLjY03iFZkBWaXmJmam5ydnm6cYZKjpKWmp6avo5h8ra6voKq6mKF1tbC2Cb OzoLquv7C7w7G0xcbEzGe6xciZj8+BP5DO2sNklF6Ch9g63NqJoN9ZXFB95dQ15ufu6Zrg5p7a5D Fg8Nz1UAADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0271_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0272.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
Example B: Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x =3D 0, –3 and 3.  x =3D 0 and 3 are odd-ordered roots and root x =3D –3 is an even-ordered root.  Draw a l= ine, mark off these roots and their types.
x=3D0 (odd)
x=3D3 (odd)
x=3D-3 (even)
Sample a point, say x =3D 4 and we get P(4) positive. = Mark the segment as such.  By the theorem, across the root x =3D 3 the sign change= s to "–" because x =3D 3 is odd-ordered.  Similarly, acro= ss the root x =3D 0, the sign changes again to "+".  But across x = =3D -3 the sign stays as "+" because it even-ordered and the chart is completed.
change
sign
change
sign
+
sign
unchanged
+
+
x=3D4
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0272_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wETAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AREAgQAAAAAAALvg4/8AAALYhI+py+0Po5y02ouz3nH4D4aix5XmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9xn4BvoOsFhsRdstgYEbEE5fVy333CknE2vN+9zPZGP 52cE6IaVN0UYKAiUeLjY03iFZkBWaXmJmam5ydnm6cYZKjpKWmp6avo5h8ra6voKq6mKF1tbC2Cb OzoLquv7C7w7G0xcbEzGe6xciZj8+BP5DO2sNklF6Ch9g63NqJoN9ZXFB95dQ15ufu6Zrg5p7a5D Fg8Nz1UAADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0272_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AAUAgAAAAAAAAAICjF0AO1== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0243.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0286.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0286_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0287.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
Factor the expression:
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0287_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0276.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
Factor the expression:
2x2 – x3 
x2 – 2x + 1
=3D
x2(2 – x)
(x – 1)2
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0276_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0276_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0276_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0277.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
Factor the expression:
2x2 – x3 
x2 – 2x + 1
=3D
x2(2 – x)
(x – 1)2
Roots of the numerator are x =3D 0 (even-ordered)
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0277_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0277_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0277_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0278.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
Factor the expression:
2x2 – x3 
x2 – 2x + 1
=3D
x2(2 – x)
(x – 1)2
Roots of the numerator are x =3D 0 (even-ordered) <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>and x =3D 2 (odd-ordered).
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0278_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0278_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0278_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0279.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
Factor the expression:
2x2 – x3 
x2 – 2x + 1
=3D
x2(2 – x)
(x – 1)2
Roots of the numerator are x =3D 0 (even-ordered) <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>and x =3D 2 (odd-ordered). The root of the denomina= tor is x =3D 1 (even-ordered).
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0279_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0279_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0279_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0280.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1
< 0
Factor the expression:
2x2 – x3 
x2 – 2x + 1
=3D
x2(2 – x)
(x – 1)2
Roots of the numerator are x =3D 0 (even-ordered) <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>and x =3D 2 (odd-ordered). The root of the denomina= tor is x =3D 1 (even-ordered).  Draw= and <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>test x =3D 3, we get " – ".&nb= sp;
x=3D0 (even)
x=3D2 (odd)
x=3D 1 (even)
x=3D3
------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0280_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0280_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBv AAEAgAAAAAAAAAIJjI+py+0Po1QFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0280_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBw AAEAgAAAAAAAAAIJjI+py+0Po1wFADs= ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0280_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wETAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAADT AREAgQAAAAAAALvg4/8AAALdhI+py+0Po5y02ouz3rz7DzLDSJbmOYTqyrbuC8fyTK/ojdb6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9yu9wsOi8fkslkVSKfPNPWa7XK/4fRLQIDHB+rxvH7P 93HnJwAYeNgwSGiI2KHox9iI8ZgXaTUnOUmol+m4ydlpQVnZY7nhZiCnusra6voKGys7q/pZSIub q7vLG2vbCxwsPPw7bHwcSjFamKyxzNw88WwqQ52BGi2xbJ3tsN0t/cn9Mi5aDn5AeY6eoL4OnrbI zic3b65mr40dVgAAOw== ------=_NextPart_01C82E62.49A7DE30 Content-Location: file:///C:/D88A1D69/5SignChartsandInequalitiesII_files/slide0281.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Sign Charts and Inequalities II
The theorem may be generalized to rational formulas
that is factorable, th= at is, both the numerator and the denominator are factor= able.
Example B: Solve the inequality
2x2 – x3 
x2 – 2x + 1