ให p, q, r และ s เป นพห นามใดๆ จะได ว า

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1 เศษส วนของพห นาม

2 ให A และ B เป นพห นาม พห นามใดๆ โดยท B 0 เร ยก B A ว า เศษส วนของพห นาม การดาเน นการของเศษส วนของพห นาม ให p, q, r และ s เป นพห นามใดๆ จะได ว า Q P R Q P Q R Q P R Q P Q R R Q P S P Q R S Q P R S R S Q P หล กการแก สมการเศษส วนของพห นาม ค อ... ต องพยายามทาส วนให หมดไป และควรทาการตรวจสอบค าของ ต วแปรท ได ว า ทาให สมการเป นจร งหร อไม

3 ข อ 1 ผลล พธ ของ Sol n 4 m n m n m n จากโจทย จะได m n 4 m n 4 4 n m n m n n 4 4 m n m n m n m n 1 m n 4 m n m = 1 m (m 1 m 1 )(n ) mn mn 1 n 4 m n n m n 4 4 m n 1 m n m n m m n 4 m n 4 m n 1 m n เท าก บเท าใด ( m m 1 )( n n 1 ) m mn n 1 1 m n m n 4 m n m 4 m n m n 1 4 n m n 1

4 ข อ ร ปอย างง ายของ a a b c c b ab ac ค อเท าใด Sol n a b c ab จาก a c b ac = (a + b) c (a + c) b เห นไหมว า... ม นอย ในร ป ผตกลส. = = (a + b c)(a + b + c) (a + c b)(a + c + b) abc abc

5 ถ า = 1, y = 7 ค าของ ม ค าเท าก บเท าใด ข อ 3 y y y y y y 3 3 Sol n จาก = = = y y y y y y 3 3 y y ) y y y)( ( y y) y)( ( y y 1 7

6 ข อ 4 จงหาค าของ Sol n จาก = = 1 ( ) = 1 = 0

7 ผลสาเร จของ ม ค าเท าก บเท าใด ข อ 5 a b b a Sol n จาก = = = = = 1 a b b a a b b a a b a b a b 1 1 b a a a b b a b a b

8 ข อ 6 จงหาค าของ + 1 เม อ Sol n จะได ( ) 7 (1 ) 7 (1 ) 3 7 ( ) 7 1 น า 7 หารตลอด จะได = ( ) ( 3) ( ) ( 1) ( 3)( ) = ( 1)( ) 1 1 ( 3)( ) = ( 1)( ) = = = 3

9 ร ปอย างง ายของ เท าก บเท าใด ข อ n m n m Sol n จาก = = = n m n m 4 4 n m n m n m ) n )(m n (m n 1 m 1

10 ข อ 8 ร ปอย างง าย ba ab b ab ab a ab ab เท าก บข อใด Sol n จาก ba ab b = = = ab ab a ab ab a(a b)(b a) b(a+b)(a+b) + (a+b)(a b)(a b) ab(a b)(a + b) (3a b a 3 ab ) (a b+3ab +b 3 )+ (a 3 a b ab +b 3 ) ab(a b)(a + b) 6ab ab(a b)(a + b) = 6b b a

11 ก บ คร มงคล วงศ พย คฆ เสร มความร ว ชาคณ ตศาส โรงเร ยนม ธยมว ดส งห ง... พห นามและเศษส วนของพห นาม

12 นาม = โดยท วไป การแยกต วประกอบของพห นามใด ค อ การเข ยนพห นามน นในร ปการค ณของพห นามท ม ด กร ต ากว า 5 ( + 3) ( + + 1) ต อไปน เป นต วอย างการแยกต วประกอบของ 3 ( + + 5) พห นามโดยใช สมบ ต การแจกแจง

13 4 4 ให น กเร ยนใช สมบ ต การแจกแจงแยกต วประกอบ ของพห นามต อไปน =. 6 3 = 3..s 3 + s = 4. 5t 15t = 5..z 3 z + 9z = 6. 3u 6u 3 7u = 4 ( + 4) 3 ( ) s (s + ) 5t (t 3) z (z z + 9) 3u (u u 9) y ( y + )

14 พห นามด กร สองต วแปรเด ยว ค อ พห นามท เข ยน ได ในร ป a + b + cเม อ a, b, cเป นค าคงต ว ท a 0 และ เป นต วแปร การแยกต วประกอบของพห นาม a + b + cเม อ a, b, c เป นค าคงต ว ท a 0 เป นการแยกต วประกอบโดย เข ยนพห นามด กร สองน นในร ปการค ณของพห

15 ในกรณ ท c = 0 เราสามารถแยกต วประกอบของพห นามด กร สองโดยใช สมบ ต การแจกแจงใน ล กษณะเด ยวก บท กล าวมาแล ว ให น กเร ยนใช ว ธ ( 4) ด งกล าวแยกต วประกอบของพห นามต อไปน ( + 9) = = = 4..a a = 3 ( 3) a ( ) (a + b)

16 = + (5 + ) + 10 ในกรณ ท a = 1 b และ cเป นจานวนเต ม และ c 0 พห นามด กร สองจะอย ในร ป + b + cเราสามารถ แยกต วประกอบของพห นามด งกล าวได โดย อาศ ยแนวค ดด งน ( + 5)( + ) = ( + 5)() + ( + 5)() = ( + 5) + ( + 10) = + (5 + ) + 10

17 จากว ธ หาผลค ณของ ( + 5)( + ) ในข อ 1 จะได ข นตอนการแยกต วประกอบของพห นามของ ด งน = + (5 + ) + 10 = + (5 + ) + 10 = ( + 5) + ( + 10) = ( + 5)() + ( + 5)() = ( + 5)( + )

18 จงใช ว ธ การข างต นแยกต วประกอบของพห นาม ต อไปน = = = = = = ( + 4)( + 3) ( + )( + 3) ( + 3)( + 5) ( + )( + 7) ( + 1)( + 14) ( + )( + 11) ( + a)( + b)

19 สร ป การแยกต วประกอบของพห นามด กร สอง + b + c เม อ b และ cเป นจานวนเต ม ทาได เม อสามารถหา จานวนเต มสองจานวนท ค ณก นได c และบวกก น ได b ให d และ e แทนจานวนเต มสองจานวนด งกล าว

20 ต อไปจะพ จารณาการแยกต วประกอบของพห นามด กร สองต วแปรเด ยวในร ป a + b + cเม อ a, b, c เป นค าคงต ว ท a 0 และ c 0 เพ อความสะดวก เราจะเร ยก a ว าพจน หน า b ว าพจน กลาง และ c ว าพจน ท าย ให น กเร ยนพ จารณาการค ณของพห นามหน ง โดย ใช สมบ ต การแจกแจงต อไปน

21 จากการค ณข างต น เราอาจเข ยนแผนภาพแสดง 6 ว ธ หาพห นามท เป นผลล พธ ได ด งน 1. (3 5)( + 1) พจน หน าของพห นามในวงเล บแรกค ณก บ 5 พจน หน าของพห นามในวงเล บหล ง ได พจน หน า ของพห นามท เป นผลล พธ

22 3 3. (3 5)( + 1) 10 ผลบวกของผลค ณระหว างพจน หน าของพห นามในวงเล บแรกก บพจน หล งของพห นามใน วงเล บหล ง และพจน หล งของพห นามในวงเล บ แรกก บพจน หน าของพห นามในวงเล บหล ง ได

23 ด งน นในการแยกต วประกอบของพห นามของ ทาได ด งน 1. หาพห นามด กร หน งสองพห นามท ค ณก นแล ว ได 6 ซ งอาจเป น 3 ก บ หร อ 6 ก บ เข ยนสอง พห นามน นเป นพจน หน าของพห นามสองพห นามด งน

24 . หาจานวนสองจานวนท ค ณก นแล วได 5 ซ ง อาจเป น 5 ก บ ( 1) หร อ ( 5) ก บ 1 เข ยนจานวนท ง สองเป นพจน หล งของพห นามในข อ 1. ด งน (3 + 5)( 1) หร อ (6 + 5)( 1) (3 1)( + 5) หร อ (6 1)( + 5) (3 5)( + 1) หร อ (6 5)( + 1)

25 3. หาพจน กลางของพห นามท เป นผลล พธ จากผล 3 ค ณของพห นามแต ละค ในข อ. ด งน (3 + 5)( 1) (3 5)( + 1) 10 ได พจน กลางเป น ได พจน กลางเป น 7 15 (3 1)( + 5) (3 + 1)( 5) ได พจน กลางเป น 13 ได พจน กลางเป น 13

26 6 (6 + 5)( 1) (6 5)( + 1) 6 5 ได พจน กลางเป น 30 5 ได พจน กลางเป น 30 (6 1)( + 5) (6 + 1)( 5) ได พจน กลางเป น 9 ได พจน กลางเป น 9

27 จะเห นว า ผลค ณของ 3 5 ก บ + 1 ได พจน กลาง เป น 7 ด งน น (3 5)( + 1) พห นาม แยกต วประกอบได เป น ให น กเร ยนแยกต วประกอบของพห นามต อไปน (3 + 5)(4 7) (3a + 4)(a + 3) (3y 1)(3y 1) 1. 6a + 17a + 1 = (6 m)(1 m). 9y 6y + 1 = (5 3n)(7 n)

28 square) ร ปท วไปของพห นามท เป นกาล งสองสมบ รณ ค อ A + AB + B และ A AB + B เม อ A และ B เป นพห นาม แยกต วประกอบได ด งน A + AB + B = (A + B) A AB + B = (A B)

29 ร ปท วไปของพห นามท อย ในร ปผลต างของกาล ง สอง ค อ A B เม อ A และ B เป นพห นาม แยกต ว ประกอบได ด งน A B = (A + B)(A B) two square)

30 ให น กเร ยนแยกต วประกอบของพห นามต อไปน = = y + 5y = y + 5y = = 6. ( 5) 49y = 7..p (q r) = ( + 3) ( 3) (3 + 5y) (6 5y) (8 15)(8 + 15) ( 5 7y)( 5 + 7y) (p q + r)(p + q r)

31 ทาเป นกาล งสองสมบ รณ การแยกต วประกอบของพห นามด กร สอง + b + c โดยว ธ ทาเป นกาล งสองสมบ รณ สร ปเป นข นตอน ด งน 1. จ ดพห นามท กาหนดให อย ในร ป + p + c หร อ p + c เม อ p เป นจานวนจร งบวก

32 + p + c = ( + p + p) p + c = ( + p) (p c). p + c = ( p + p ) p + c = ( p) (p c). 3. ถ า p c = d เม อ d เป นจานวนจร งบวก จากข อ. จะได + p + c = ( + p) d. p + c = ( p) d.

33

34

35 นามด กร ส งกว าสอง เม อ A และ B เป นพห นาม ผลบวกของกาล งสาม A 3 + B 3 = (A + B)(A AB + B ) ผลต างของกาล งสาม A 3 B 3 = (A B )(A + AB + B )

36 พห นาม ฤษฎ บทเศษเหล อ (Remainder Theorem) ษฎ บทต วประกอบ (Factor Theorem) c โดยท c เป นค าคงต ว เศษท เหล อจากการ ถ า P(c) = 0 แล ว จะได c c หารเท าก บ P(c) เป นต วประกอบหน ง

37 ต วอย างข อสอบแข งข น เร อง

38 1. จงแยกต วประกอบของพห นามต อไปน (1) a + a(c 3b) + b bc = a + ac 3ab + b bc = a 3ab + b + ac bc = (a b)(a b) + (a b)c = (a b)(a b + c)

39 1. จงแยกต วประกอบของพห นามต อไปน () ( + 1)( + )( 1)( ) 40 = ( 1)( 4) 40 = = ( 9)( + 4) = ( 3)( + 3)( + 4)

40 1. จงแยกต วประกอบของพห นามต อไปน (3) ( + 1)( )( 3)( 6) + 0 = ( + 1)( 6)( )( 3) + 0 = ( 5 6)( 5 + 6) + 0 = ( 5) = ( 5) 16 = ( 5 + 4)( 5 4) = = [ ] ( 1)( 4) ( 5 ) ( 41 ) 4 ( 1)( 4)( 5 41 )( 5 41 )

41 . ส มประส ทธ ของ a 3 b ท เก ดจากผลค ณของพห นาม (a 5 a 4 + 3a 5ab b )(a 4 4a 3 b + a + 3ab + b ) เราจะหาเฉพาะพจน ท ค ณก นแล วได a 3 b ลอง (a 5 a 4 + 3a 5ab b )(a 4 4a 3 b + a + 3ab + b ) ด ซ เด กๆ... (3a )(3ab) + ( 5ab)(a ) = (9a 3 b) (10a 3 b) = a 3 b

42 3. ถ า (3 3 7 p 5)( +3+6) = แล ว p ม ค าเท าก บเท าใด ข อน ไม จาเป นต องหาผลค ณ ต องการค าของ p ค ณเฉพาะพจน ท ม p ก พอ (3 3 7 p 5)( +3 +6) = พ จารณาพจน ท ค ณก นได ; ( p)(6) + ( 5)(3) = 3 6p 15 = 3 6p = 18 p = 3

43 4. ให เป นจานวนเต ม และ p = (+)(+4)(+8)(+10) + n จะได ค าของ n ท เป นจานวนน บท น อยท ส ดเท าไรจ งจะ จาก p = ( + )( + 4)( + 8)( + 10) + n = ( + )( + 10) ( + 4)( + 8) + n = ( )( ) + n = ( + 1) + 5( + 1) n p เป นจานวนจร ง ถ าขวาม อจ ดเป นกาล งสอง สมบ รณ ได

44 5. กาหนด A B C ( 3) 3 ค าของ A, B, C เท าก บ น า ค.ร.น. ของต วส วนค ณตลอด จะได เท าใด = A(+3) + B( 4)(+3) + C( 4) = A( +6+9) + B( 1) + C( 4) = (A+B) + (6A B+C) + (9A 1B 4C)

45 เท ยบส มประส ทธ ; A + B = 1 6A B + C = 16 9A 1B 4C = 18 น กเร ยนลองแก ระบบสมการเลยคร บ... จะได ว า A =, B = 1, C = 3

46 6. กาหนดให p( + ) = จงหาค าของ p() ให + = k จะได = k ด งน น p(k) = 3(k ) + 10(k ) + 3 = 3(k 4k + 4) + 10(k ) + 3 = 3k k 5 น นค อ p() = 3 5

47 7. ถ า และ aต างหารด วย + เหล อเศษเท าก น จงหาค า a ใช ทบ.เศษเหล อ จะได ( ) 3 + ( ) + 3( ) + = ( ) 3 + ( ) + a a = 4

48 (1 ) ( ) ผลสาเร จ ม ค าเท าใด (1 ) ( ) (1 )(1 4 ) (4 ) 5 ( )( ) (1 )(1 ) 1 4 (1 )(1 4 ) ( )( ) ( )(1 ) ( )( ) (1 )(1 ) 1 4 ( )

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