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B@   0 " (    C AUC:\Documents and Settings\harrisonpang\My Documents\powerpoint\icons\ppt_ch7_icon.jpg"x9  0P1Ef @D9!0!0 m7.2 Division of Polynomials"(2dff  #  0H6E  @K9!0!0 {7.3 Remainder Theorem dff   @`   `A ? ?"TPL !0!0   <BE k@ VContents e    0$GE?"@ O7 2bH    0dJE?"?@$ hMore about Polynomials" 2f,  ;  6NEL l @U9!0!0 '7.4 Factor Theorem and Its Applications ((dff (  @`  C AUC:\Documents and Settings\harrisonpang\My Documents\powerpoint\icons\ppt_ch7_icon.jpg"p$  S ,AXQuit_C"@ ;$ @3 !0!0$  HWE?"C D @3 !0!0 VHome"g    0\E@=9!0!0 v$7.1 Polynomials and Their Operations"%(2%dff % |  dAW"޽h ?C:\4B Page 1.jpg ̙330  @k$0  (  $ j$ Bo?"p0 ,$D0  $ 6Q> ,$D 0 pA. Polynomials in One Variable" 2fff    Q$ c $U><$ 0  > r&7.1 Polynomials and Their Operations''G '  R$  `A  ? ?"40L   !0!0 S$  `A  ? ?"TPL   !0!0H $ 6Y>t,$D 0 >In general, a polynomial in one variable x can be expressed as@?)cbc ?  U$ s *AQ ;?? 8 ;$D 0 Y$ s *AR ;??S  8 ;$D  0 c$ BAV 3??p8 3$D 0 g$ Txo> gjJ?" t$D  0@8___PPT9 A polynomial with a non-zero constant term a0only is of degree 0. A polynomial consisting of 0 only is called the zero polynomial and has no degree.  2,    @` h$  fA` 3? gjJ?"P8 3$D 0 i$ Tp> gjJ?"  t$D  0@8___PPT9 BA polynomial in one variable x can be regarded as a function of x.nC 2! C  @` k$ <$}>| KP ,$D 0 nRemarks:6  2gffeff  H $ 0޽h ? ̙33  ___PPT10| +*GD' = @B D;' = @BA?%,( < +O%,( < +Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q$%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*Q$D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*Q$D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*Q$D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*Q$D' =%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-}6B!slide(fromLeft)*<3<*$D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*b$%(D' =-6B)barn(outHorizontal)*<3<*b$D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*\$%(D' =-6B)barn(outHorizontal)*<3<*\$DG' =%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^$%(D' =-}6B!slide(fromLeft)*<3<*^$DA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*_$%(D' =-}6B!slide(fromLeft)*<3<*_$D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*]$%(D' =-s6Bwipe(left)*<3<*]$D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*`$%(D' =-6B)barn(outHorizontal)*<3<*`$D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*a$%(D' =-6B)barn(outHorizontal)*<3<*a$++0+$K  ++0+Q$  ++0+]$\ ++0+^$ +   B@ Z R  (  .   Zb- gjJ?",$ 0 b Example 7.1T& (2 gff  4   6-"` ,$@ 0 :B. Addition, Subtraction and Multiplication of Polynomials"; 2;fff ;    s *A B???0 8 B$D 0   c $A\ B??h X 8 B$D 0   BAV 3??<38 3$D 0   fA 3? gjJ?"   8 3$D  0  <-@ ,$D 0 q Solution:8 gffgff     `A 3? ?"40L 33!0!0   `A 3? ?"TPL 33!0!0z  7    0 ,$D 0z  JAKffVv} 0   0X 7  3Group the like terms to calculate the coefficients. 4024b 4   C AC:\Documents and Settings\Harrison\My Documents\tips_for_students_4B.gif  ,$D 0r  S    H  0޽h ? ̙33{    P R, (  , F, c $>  > ~&7.1 Polynomials and Their Operations''G&   G,  `A  ? ?"40L   !0!0 H,  `A  ? ?"TPL   !0!0 I, c $A 4??`P 8 4$D 0 J, c $A "4??p ` 8 "4$D 0. K, Z> gjJ?"P$,$ 0 b Example 7.2T& (2 gff   L,  fAb '4? gjJ?"  8 '4$D 0 M, <ؔ>,$D 0 q Solution:8 gffgff  z  7  O, 0,$D 0z P, JAKffVv} 0  Q, 0\> 7  0@Subtract A from B means B  A.0202 bbbbbbbbbb !  R, C AC:\Documents and Settings\Harrison\My Documents\tips_for_students_4B.gif@Rr^,$D 0H , 0޽h ? ̙33 x ___PPT10X +D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2, %(D' =-s6Bwipe(left)*<3<*2, D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*I,%(D' =-6B)barn(outHorizontal)*<3<*I,D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*J,%(D' =-6B)barn(outHorizontal)*<3<*J,+8+0+2,t  +A* B@ ` E@( (  @  @ 6̴> ,$D 0 lC. Equality of Polynomials" 2fff   +@ c $>  > r&7.1 Polynomials and Their Operations''G '  -@  `A  ? ?"40L   !0!0 .@  `A  ? ?"TPL   !0!0 0@ T > gjJ? "*,$D 0 Two polynomials f (x) and g(x) are equal for all real values of x if the coefficients of like powers of x in the two polynomials are equal.  2cbbbbcbbbb"cb&cb#cb   1@ s *A d4?? P 8 d4$D 0 @@ BAV g4??<8 g4$D 0l @ s  E@ @s ,$D 0 A@ V>AKffVv}@ s  N&(2gff   B@ 0l>m <  HDf (x) g(x) means that f (x) and g(x) are equal for all real values of x. Also, we can say that f (x) is identical to g(x), or f (x) g(x) is an identity. 02bbbb bbbbb"bbbbbbbbbbbbbbb   D@ C AC:\Documents and Settings\Harrison\My Documents\tips_for_students_4B.gif0 ,$D 0H @ 0޽h ? ̙33 ___PPT10+WDa' = @B D' = @BA?%,( < +O%,( < +DG' =%(D' =%(DA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*/@%(D' =-}6B!slide(fromLeft)*<3<*/@DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =-}6B!slide(fromLeft)*<3<* @D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*=@%(D' =-6B)barn(outHorizontal)*<3<*=@D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:@%(D' =-s6Bwipe(left)*<3<*:@D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*;@%(D' =-6B)barn(outHorizontal)*<3<*;@D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*<@%(D' =-6B)barn(outHorizontal)*<3<*<@D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*6@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*6@D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>@%(D' =-s6Bwipe(left)*<3<*>@+p+0+ @t  ++0+:@t  +  B@   p cL S (  L RL  `A  ? ?"40L   !0!0 SL  `A  ? ?"TPL   !0!0> WL 0> N ,$D 0 ZWhen a polynomial is divided by a divisor, we get a quotient and a remainder. We can write [ 2[c [ ` ZL c $A m1?? l  m1 YL s *A 4??p 8 4$D 0 ^L S ><$D 0  > h7.2 Division of PolynomialsG   _L 6| >o,$D 0 aA. By Monomials" 2fff   `L BAV 4??p8 4$D 0 bL B$> ?"   ,$D 0 where the remainder is zero or the degree of the remainder is less than that of the divisor. It is known as the Division algorithm of polynomials.2 2pc#c   cL N)> ?" f ,$D 0 Rdividend = divisor quotient + remainderF* 2cc * H L 0޽h ? ̙33___PPT10+܏PD' = @B D' = @BA?%,( < +O%,( < +Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*XL%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*XLD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*XLD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*XLD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*XLDG' =%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*UL%(D' =-}6B!slide(fromLeft)*<3<*ULDA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*TL%(D' =-}6B!slide(fromLeft)*<3<*TLD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*\L%(D' =-6B)barn(outHorizontal)*<3<*\LD' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*WL%(D' =-6B)barn(outHorizontal)*<3<*WL++0+ULt  ++0+WLt  ++0+XLt  +' B@ VN8 (    7  Bo?"p  @@,$D 0 5  Bo?" @ ,$D0   c $,5>  > 7.2 Division of Polynomials@GG G      `A  ? ?"40L   !0!0    `A  ? ?"TPL   !0!0% !  Zd;> gjJ?" 0O,$@ 0 YStep 1:"(2c  % "  Z$,> gjJ?" 0O ,$@ 0 YStep 2:"(2c   ,  c $A 4??8n ` 8 4$D 0 -  c $A 4??8 v 8 4$D  0 .  67 ,$D 0 hB. By Linear Binomials" 2fff   /  BAV 4??p8 4$D 0 3  c $A& 4??x 08 4$D 0 4  c $A( ;??@  8 ;$D 0 8  N7 gjJ?"  t ,$  0 $2x2  4x is obtained by multiplying x  2 and 2x. Then 2x2  4x is subtracted from the dividend 2x2 + 3x  4 to give 7x  4. 0P2bbjbbcbbcbbcbbjbbbcbbjbbbcbb b  H   0޽h ? ̙33___PPT10+njD' = @B D' = @BA?%,( < +O%,( < +DG' =%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-}6B!slide(fromLeft)*<3<* DA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-}6B!slide(fromLeft)*<3<* DW' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*! %(D' =-6B)barn(outHorizontal)*<3<*! DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =-6B)barn(outHorizontal)*<3<*, D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*' %(D' =-6B)barn(outHorizontal)*<3<*' DW' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*" %(D' =-6B)barn(outHorizontal)*<3<*" DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*- %(D' =-6B)barn(outHorizontal)*<3<*- D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*+ %(D' =-6B)barn(outHorizontal)*<3<*+ ++0+ t  ++0+! t  ++0+" K  +$& B@ ^ V  fp (   f Bo?"`  ,$D0 e Bo?" p,$D0 V  `A  ? ?"40L   !0!0 W  `A  ? ?"TPL   !0!0% Y Z(#7 gjJ?"wT,$ 0 YStep 3:"(2c  % Z ZH'7 gjJ?"L t ,$ 0 YStep 4:"(2c   \ N77 gjJ?" V ,$ 0 7x  14 is obtained by multiplying x  2 by 7. Then 7x  14 is subtracted from 7x  4 and the remainder is 10. We stop here since the degree of the remainder is less than the degree of the divisor.J 2bbbcbbcbcbbbcbbbcbYc   ^ c $A 4??F ,8 4$D 0 a c $A 4??4 h0 L 8 4$@ 0 b c $A$ ;?? tT8 ;$@  0 c S 7  7 h7.2 Division of PolynomialsG   d c $A# <??` (8 <$D 0H  0޽h ? ̙33VN___PPT10.+VD2' = @B D' = @BA?%,( < +O%,( < +DW' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-6B)barn(outHorizontal)*<3<*YDI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-6B)barn(outHorizontal)*<3<*^D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =-6B)barn(outHorizontal)*<3<*`DW' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D' =-6B)barn(outHorizontal)*<3<*aDV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D' =-6B)barn(outHorizontal)*<3<*ZD' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B)barn(outHorizontal)*<3<*\DW' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' =-6B)barn(outHorizontal)*<3<*bDV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =-6B)barn(outHorizontal)*<3<*]++0+Yt  ++0+Z ++0+\ ++0+] +#0 B@ 0P>(  l `pR ,p`R,$D 0  PAKffVv}`pR,$@  0  68N7 r  To start with the method of long division, we need to rearrange the terms of the dividend and the divisor in descending powers of x.P 2bbbb  @    `A  ? ?"40L   !0!0    `A  ? ?"TPL   !0!0  6[7E,$D 0 ` Example 7.7T$  2 gff    s *A! s8??h 8 s8$D 0  c $A* 4??p  8 4$@ 0  6_7 ,$D 0 mC. By Quadratic Polynomials" 2fff    BAV 4??<8 4$D 0 " TA- 4? ?"t  , 8 4$D  0l 0p p  $p  p ,$D  0 ! NA) 4? ?"0  p  4 # NA' 4? ?"7p 4 % NA, 4? ?" 0 P 8 4$D  0l b4*  )p  ,$D  0   TA% 4? ?"D4 4 ' NA0 4? ?"b * 8 4$D  0 * <7\0 ,$D 0 q Solution:8 gffgff  o + B7 ?"0 K,$D  0 ]The procedure stops when the degree of the remainder is less than the degree of the divisor.  ^ 2^b ^ @ - c $A+ P5?? h 8 P5$D 0 . C AC:\Documents and Settings\Harrison\My Documents\tips_for_students_4B.gif0P,$D 0 / S XK7  7 p7.2 Division of Polynomials$G  H  0޽h ? ̙33___PPT10+*()jD' = @B D' = @BA?%,( < +O%,( < +DG' =%(D' =%(DN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-}6B!slide(fromLeft)*<3<* DA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-}6B!slide(fromLeft)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*Du ' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*++0+  ++0+ ++0+t  + B@ <4\\p(  \ (\  `A  ? ?"40L   !0!0 )\  `A  ? ?"TPL   !0!0 D\ 6ء7,$D 0 ` Example 7.8T$  2 gff   E\ c $A1 <??8 <$D 0 F\ c $Ar 15?? 8 15$@ 0 G\ NAz 95? ?" @ 8 95$D 0l v  Z\v ,$D 0 I\ ZA6 H5? ?"p v  H5 J\ ZA' J5? ?"O J5 K\ NA3 ;5? ?" 8 8 ;5$D 0l @ k  R\@ k ,$D 0 M\ TA} C5? ?"p(  C5 N\ TA{ 75? ?" @ k 8 75$D  0 O\ < vK,$D 0 q Solution:8 gffgff   P\ c $A ;??X , 8 ;$D  0Ql `p  Y\p` ,$D  0 T\ PAKffVv}`p ,$@  0 U\ 6|K R  gSince the remainder is 0, the polynomial 2x3 + 7x2 + 7x + 5 is divisible by 1 + x + x2.h 2:bbjbbjbbb bbbbbjb h  V\ C AC:\Documents and Settings\Harrison\My Documents\tips_for_students_4B.gif0P,$D  0 [\ S \K  K h7.2 Division of PolynomialsG  H \ 0޽h ? ̙33  ___PPT10 +DF ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D(' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<**\%(D' =-s6Bwipe(left)*<3<**\D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*4\%(D' =-6B)barn(outHorizontal)*<3<*4\DW' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*5\%(D' =-6B)barn(outHorizontal)*<3<*5\DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*6\%(D' =-6B)barn(outHorizontal)*<3<*6\+p+0+*\2 ++0+6\ +(    3d (  d 0d Bo?" ,$D 0 d  `A  ? ?"40L   !0!0 d  `A  ? ?"TPL   !0!0 $d Z6 gjJ?"6 T,$ 0 "By putting x = a into (*), we have# 2 cbcbbcb c #  %d0 NA: ;? ?\ , H ;$@ 0  'd0 NA< 1? ? Y H 1$D 0 < *d <B ?"P`X,$D 0 $When a polynomial P(x) is divided by x  a, a quotient Q(x) and a remainder R is obtained. P(x) can be expressed in terms of x  a, Q(x) and R as . 2cbbbcbbb cbbbbcbcbbbbcbbbcbbbbcbbc   +d c $A] ;??08 ;$D 0 -d S C<$D 0    /d Z< gjJ?" ,$ 0 When a polynomial P(x) is divided by x  a, the remainder R is equal to P(a).O 2cbbbbcbcbcbbbbc O # 1d N  ?"0 `  ,$D  0 cRemainder Theorem"(2c   3d C DA,A:\Non-Foundation.gif,$D 0H d 0޽h ? ̙33___PPT10.+3D' = @B D' = @BA?%,( < +O%,( < +Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!d%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*!dD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*!dD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*!dD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*!dD' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$d%(D' =-6B)barn(outHorizontal)*<3<*$dDJ' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*%d%(D' =-6B)barn(outHorizontal)*<3<*%dDI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*'d%(D' =-6B)barn(outHorizontal)*<3<*'dD+' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*"d%(D' =-o6Bdissolve*<3<*"dD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#d%(D' =-o6Bdissolve*<3<*#d++0+!dt  ++0+#d ++0+$dt  +f   B@   (  r  S pD      s *A 5? ?p@ 8 5$@ 0   s *A 5? ? 8 5$D 0   <ԯ ?"P`,$D 0 FWhen a polynomial P(x) is divided by px  q, the quotient Q(x) and the remainder R are obtained. Then we haven 2cbbbcbbbcbbbbcbc2%  F     c $A4 ;??0L 8 ;$D 0   c $A9 5?? x8 5$D 0   `A 5? ?"40L 55!0!0   `A 5? ?"TPL 55!0!0  C DA,A:\Non-Foundation.gif,$0H  0޽h ? ̙33t2  )!2p0 (  p +p Bo?" P,$D 0, p 62KP{,$D 0 If P(x) is a polynomial and P(a) = 0, then x  a is a factor of P(x). Conversely, if x  a is a factor of a polynomial P(x), then P(a) = 0.40 2cbbbbcbbbbcbcbcbbbbcbcbbbbbcbbbbc   @`  p c $D$K<$ 0  K s'7.4 Factor Theorem and Its Applications((G (  p  `A  ? ?"40L   !0!0 p  `A  ? ?"TPL   !0!0 )p 6(K' ,$D 0 cA. Factor Theorem" 2fff   *p BAV 6??<8 6$D 0l P`   0p` P ,$D 0 p  `8.K gjJ?"P`   .If P(x) is a polynomial and = 0, then px  q is a factor of P(x). Conversely, if px  q is a factor of a polynomial P(x), then = 0.P02cbcbcbcbbbbbcbcbcbc@0  )  ; n $p s *A <??R `  <n ,p s *A> <?? `  <7 .p N +- ?"P,$D 0 wFactor Theorem 80(2cc   2p C DA,A:\Non-Foundation.gif,$D 0H p 0޽h ? ̙33  ___PPT10 +{IVD' = @B D:' = @BA?%,( < +O%,( < +Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*pD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*pD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*pD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*pDG' =%(D' =%(DA' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-}6B!slide(fromLeft)*<3<*pDN' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-}6B!slide(fromLeft)*<3<*pD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-o6Bdissolve*<3<*pD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*(p%(D' =-6B)barn(outHorizontal)*<3<*(pD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =-s6Bwipe(left)*<3<* pD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*&p%(D' =-6B)barn(outHorizontal)*<3<*&pD' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*#p%(D' =-6B)barn(outHorizontal)*<3<*#pD' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*'p%(D' =-6B)barn(outHorizontal)*<3<*'p++0+p ++0+p ++0+ p  ++0+#p  ++0+'p  + B@ 6@(   ! c $4-  - '7.4 Factor Theorem and Its Applications((G'   &  `A  ? ?"40L   !0!0 '  `A  ? ?"TPL   !0!0 + Z\9- gjJ?"P t$ 0@8___PPT9 -Factorize P(x). Solve the equation P(x) = 0. . 2 cbbbbbcbbbbcbcbc .  @` , s *AA 68??Xx 8 68$D 0 . 6h?-@`,$D 0 a Example 7.12T$ 2gff   / s *Af ;? ? || < 8 ;$@ 0 0 s *A 88? ?8 88$@  0 1 ZlD- gjJ?"  ,$  0 :\ x + 1 is a factor of P(x). 2bbcbbbb   2 Z- gjJ?" 0A ,$ 0 S(a)  2c   3 <{- `[ ,$D 0 q Solution:8 gffgff   4 6- ,$D 0 w%B. Applications of the Factor Theorem"& 2&fff &  5 BAV a9??<8 a9$D 0 6 C DA,A:\Non-Foundation.gif,$0H  0޽h ? ̙33  ___PPT10 +&fD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D(' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-s6Bwipe(left)*<3<*)D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B)barn(outHorizontal)*<3<*-D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-6B)barn(outHorizontal)*<3<*++p+0+)? ++0++  +,  B@   0 W$ , (    c $$<$0    3  `A  ? ?"40L   !0!0 >0 TA %8? ? N H %8$@ 0  @0 TA C8? ?P  H C8$@ 0 7 C  f gjJ?" T ,$D 0 _(b)  2c   F NAN F8? ?" , h 8 F8$D 0 M ZT gjJ?" t$0@8___PPT9 k-Factorize P(x). Solve the equation P(x) = 0. . 2 cbbbbbcbbbbcbc .  @` N c $AA :8??x 8 :8$0 O 60v`,$0 a Example 7.12T$ 2gff   P <v\`[0 ,$0 q Solution:8 gffgff   U NAI H8? ?"p 0 8 H8$D 0 W C DA,A:\Non-Foundation.gif,$0H  0޽h ? ̙33___PPT10.+DS' = @B D' = @BA?%,( < +O%,( < +Dy' =%(%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* D' =%(D]' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-6B)barn(outHorizontal)*<3<*8DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-6B)barn(outHorizontal)*<3<*:DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =-6B)barn(outHorizontal)*<3<*BDW' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-6B)barn(outHorizontal)*<3<*<DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-6B)barn(outHorizontal)*<3<*;D' =%(DP' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-6B)barn(outHorizontal)*<3<*>DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B)barn(outHorizontal)*<3<*@DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-6B)barn(outHorizontal)*<3<*C++0+ } ++0+: ++0+; ++0+B ++0+C +0 P>(  X  C     S  S S  @  S @` H  0޽h ? ̙33 ,xZ[lT]c?114LK`\ "@@! 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