เอกสารประกอบการสอนเอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพิ่มเติม ม.4 เทอม 1 เร่ือง จำนวนจรงิ 1
วิชาคณติ ศาสตร3เพิ่มเติม
ม.4 เทอม 1
เรื่อง จำนวนจริง
ช่ือ …………………………………………………………………………ชนั้ ………………….เลขที่ ………….
โรง เรียน …………………………………………… ครูผ7สู อน ………………………………………………….
เอกสารประกอบการสอนวชิ าคณติ ศาสตรเ2 พมิ่ เตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจรงิ 2
แบบฝก< หดั 3.1
1. จงพจิ ารณาจำนวนท่กี ำหนดให2 จำนวนใดบ2างทเี่ ปน7 จำนวนนับ จำนวนเตม็ จำนวนตรรกยะหรือจำนวนอตรรกยะ
0, √2 , −22 , 3.1416, √4 + 1, .1 − (−8), √6 − 1 ,
3 7
7 12 5√26!,
22 , 0.09̇ , − 3 , −3.999, .(−1)!
จำนวนท่ี จำนวนนบั จำนวนเต็ม จำนวนตรรกยะ จำนวนอตรรกยะ
กำหนดให,
0
√32
−722
3.1416
√4 + 1
.1 − (−8)
√6 − 1
72 2
0.09̇
− 132
5√26!
−3.999
.(−1)!
2. ข2อความตอB ไปนเ้ี ปน7 จรงิ หรอื เทจ็
1) 1.010010001 เป7นจำนวนตรรกยะ …...........................................
2) 6.808808880... ไมเB ปน7 จำนวนตรรกยะ …...........................................
3) 0.797797797... เปน7 จำนวนอตรรกยะ …...........................................
4) 1 - √2 ไมBเปน7 จำนวนจริง …...........................................
5) 9"$# เปน7 จำนวนตรรกยะ …...........................................
…...........................................
6) ถา2 สามารถเขียนไดใ2 นรูปทศนยิ มซ้ำ แล2ว เป7นจำนวนตรรกยะ …...........................................
7) มจี ำนวนจรงิ ที่ √ + 1 = −1 …...........................................
8) มีจำนวนเตม็ ท่ีมากทสี่ ุดทน่ี 2อยกวาB 9
9) มีจำนวนตรรกยะทีม่ ากท่สี ดุ ทนี่ 2อยกวาB 9 …...........................................
เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพม่ิ เติม ม.4 เทอม 1 เรอ่ื ง จำนวนจรงิ 3
แบบฝ<กหดั 3.2
1. จงบอกสมบัติของจำนวนจรงิ ท่ีทำให2สมการหรอื ข2อความตBอไปนเ้ี ป7นจริง เม่ืออักษรทปี่ รากฏแทนจำนวนจรงิ ใด ๆ
1) %& (2 + 7) = (2 + 7) &%
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2) (-6)+0 = -6
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3) 1 ∙ =
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4) 7(-3) เปน7 จำนวนจริง
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5) [2 + ( + )] + 3( + ) = 2 + [( + ) + 3( + )]
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6) ( + )( + ) = ( + ) + ( + )
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7) −5 E'%(F = 1
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8) E5 × 9 %!F × 2 = 5 E9 % × 2F
!
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% %
) )
9) − + = 0
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10) 2 + ( + 1) = 2 + (1 + )
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรเ2 พ่มิ เติม ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 4
2. จงหาตวั ผกผนั การบวก และตวั ผกผนั การคูณของจำนวนตอB ไปนี้ ตวั ผกผันการคณู
! ( '+
−4, √5 , ) , − √%% , 1 − √7 , !√2 , √!,√&
จำนวนท่กี ำหนดให, ตัวผกผนั การบวก
−4
√5
2
7
5
− √11
1 − √7
!√2
−8
√2 + √3
3. จงพิจารณาวBาเซตท่ีกำหนดใหม2 สี มบัตใิ ดตBอไปนบ้ี า2 ง
สมบตั ิปดQ ของการบวก สมบัติปQดของการลบ สมบตั ิปดQ ของการคณู สมบัติปดQ ของการหาร (เม่อื ตัวหารไมเB ป7นศนู ยR)
เซตทีก่ ำหนดให, สมบตั ิ สมบัตปิ ดA สมบตั ิปAด สมบัตปิ ดA
ปAดของ ของการลบ ของการคณู ของการหาร
การบวก (ตัวหารไมG
เปHนศูนย)J
1) เซตของจำนวนนับ
2) เซตของจำนวนเต็ม
3) เซตของจำนวนคีล่ บ
4) เซตของจำนวนคBู
5) เซตของจำนวนเต็มทหี่ ารด2วย 3 ลงตัว
6) เซตของจำนวนตรรกยะ
7) {. . . , −5, 0, 5, 10}
8) {−1, −2, −3, . . . }
9) {−1, 0, 1}
10) K. . . , % , % , % , % , 1, 2, 4, 8, 16, . . . L
%" + # !
เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พิ่มเตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจริง 5
แบบฝ<กหัด 3.3
1. ให2 ( ) = 3 # + 2 ! − + 3 และ ( ) = # + & + 2 ! − 5 + 3 ถา2 ( ) = ( )
จงหา , และ
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2. ให2 ( ) = ! − 1 และ ( ) = ! − 2 + 3 จงหา
1) ( ) + ( )
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2) ( ) − ( )
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3) ( ) ( )
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3. ให2 ( ) = 3 ! + 5 − 1 และ ( ) = # − 5 ! + 7 จงหา ( ) ( )
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4. ถ2า ! − 12 − 28 = ( − )( − ) จงหา + และ
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เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พ่มิ เติม ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 6
5. ถ2า ! − 2 + 5 = ( − )! + ! เมื่อ > 0 จงหา และ
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6. จงหาพหุนามทเ่ี มอื่ หารดว2 ย ! + 3 แล2วได2ผลหารคอื ! − 1 และเศษเหลอื คอื − 1
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7. จงหาผลหารและเศษเหลือจากการหารพหนุ าม ( ) ดว2 ยพหุนาม ( ) เมอื่ กำหนดให 2
1) ( ) = 4 # − 3 & + 2 ! − 5 และ ( ) = !
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2) ( ) = & − 2 และ ( ) = ! + 2
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพิ่มเติม ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 7
3) ( ) = ( − # − & − ! − − 2 และ ( ) = + 2
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4) ( ) = ( + 1 และ ( ) = ! + 1
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5) ( ) = " + & + 1 และ ( ) = & − 1
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพ่มิ เติม ม.4 เทอม 1 เรื่อง จำนวนจรงิ 8
แบบฝก< หัด 3.4
1. กำหนด ( ) และ ดงั ตBอไปน้ี จงหาเศษเหลอื เมื่อหาร ( ) ดว2 ย −
1) ( ) = # − 3 + 5, = 2
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2) ( ) = 2 & + 7 ! − 5 − 4, = −3
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3) ( ) = 6 & + 13 ! − 4, = −2
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4) ( ) = # − 3 & + 4 ! − + 6, = 1
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พิม่ เตมิ ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 9
5) ( ) = 2 # − 5 & − ! + 3 + 1, = − !%
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2. จงแสดงวBา − 1 เปน7 ตวั ประกอบของ & − 2 ! − 5 + 6
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3. จงแสดงวBา + 1 เป7นตวั ประกอบของ & + ! + + 1
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4. จงหาคBา จากเงื่อนไขทก่ี ำหนดให 2
1) − 5 หาร & − 2 ! + 8 − ลงตวั
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพ่ิมเตมิ ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 10
2) + ! หาร 3 # − 2 & + − 1 เหลอื เศษ −1
&
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3) + หาร ! − 5 − 2 เหลอื เศษ −8
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5. จงแยกตัวประกอบของพหุนามตBอไปน ้ี
1) & − ! − 4 + 4
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2) & + ! − 8 − 12
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพ่มิ เติม ม.4 เทอม 1 เร่อื ง จำนวนจริง 11
3) # − 2 & − ! − 4 − 6
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พม่ิ เติม ม.4 เทอม 1 เร่ือง จำนวนจรงิ 12
7) # − 2 & + ! − 4 + 4
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8) # − 2 & − 13 ! + 14 + 24
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6. จงแยกตวั ประกอบของพหุนามตBอไปน ี้
1) 6 & − 11 ! + 6 − 1
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เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พมิ่ เตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 13
2) 6 & + ! − 11 − 6
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3) 8 # + 8 & + 6 ! + 4 + 1
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4) 3 # − 8 & + ! + 8 – 4
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรเ2 พ่มิ เติม ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 14
แบบฝก< หดั 3.5
1. จงหาเซตคำตอบของสมการตBอไปน ี้
1) & − 2 ! − 5 + 6 = 0
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2) & + ! − 8 − 12 = 0
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3) 1 − 3 ! + 2 & = 0
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4) 3 & − 2 ! − 7 = 2
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพม่ิ เตมิ ม.4 เทอม 1 เร่ือง จำนวนจรงิ 15
5) 6 − 13 + 4 & = 0
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6) & − 3 ! + + 2 = 0
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7) 5 − 6 = 2 & − 3 !
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8) & − ! − − 2 = 0
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพิ่มเตมิ ม.4 เทอม 1 เร่อื ง จำนวนจริง 16
9) 4 & + 13 ! + 4 − 12 = 0
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10) 2 # − 13 & + 28 ! = 23 − 6
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11) 4 # − 4 & − 9 ! + + 2
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12) 3 # − 8 & + ! + 8 = 4
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เอกสารประกอบการสอนวิชาคณิตศาสตรเ2 พม่ิ เติม ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 17
13) # − 2 & − 13 ! + 14 + 24 = 0
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2. ถา2 ผลคณู ของจำนวนค่ี 3 จำนวนท่เี รยี งตดิ กันเทาB กับ 1,287 จงหาจำนวนทีน่ 2อยทส่ี ุดในสามจำนวนน้ี
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3. สมมติวาB โยนลูกบอลข้นึ ไปในอากาศจากดาดฟVาตกึ ซึง่ สูง 12 เมตร สมการแสดงความสัมพันธR
ระหวาB งเวลา (วินาที) กบั ความสงู ของลกู บอลจากพ้ืนดิน (เมตร) คอื ( ) = 12 + 28 − 5 !
เม่อื ( ) แทนความสงู ของลกู บอลจากพืน้ ดินในวนิ าทที ี่
จงหาวาB ลูกบอลจะลอยอยใBู นอากาศนานกว่ี ินาทกี อB นตกกระทบพืน้ ดนิ เปน7 ครั้งแรก
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พิ่มเติม ม.4 เทอม 1 เร่ือง จำนวนจริง 18
แบบฝก< หัด 3.6
1. จงเขียนเศษสวB นของพหุนามในรปู ผลสำเรจ็
1) --! ' ' % %
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2) !-#"-," -' ' $ &
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพมิ่ เติม ม.4 เทอม 1 เรอ่ื ง จำนวนจรงิ 19
2. จงหาผลลพั ธใR นรปู ผลสำเรจ็
1) --" "' ' - & '- ! ∙ -"- '" '- '# "
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เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พม่ิ เตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจรงิ 20
3. จงหาผลลพั ธใR นรูปผลสำเร็จ
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พิ่มเติม ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 21
แบบฝ<กหัด 3.7
1. จงหาเซตคำตอบของสมการตอB ไปน้ี
1) (--,(%-) '(- %,)!) = (-,%)!(-,!)
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพม่ิ เตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจริง 22
4) -% − # = 1
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพมิ่ เติม ม.4 เทอม 1 เรื่อง จำนวนจรงิ 23
7) - '% ! − - = %"
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2. ในการบินระยะทาง 1,500 กโิ ลเมตร ครัง้ หนงึ่ เคร่ืองบินพบกับสภาพอากาศแปรปรวน
ทำให2ตอ2 งบนิ ชา2 ลงกวาB ปกติ พบวาB อตั ราเรว็ ลดลง 100 กิโลเมตรตBอช่วั โมง ทำให2ถึงที่หมายช2ากวาB ปกติ
10 นาที จงหาอตั ราเร็วของการบินปกต ิ
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พ่ิมเติม ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 24
แบบฝ<กหดั 3.8
กำหนดให2 และ เปน7 จำนวนจริง
1. จรงิ หรอื ไมB ถ2า > แลว2 ! > !
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ถา2 แล2ว 2% %
2. จรงิ หรอื ไมB 0 และ
≠ 0, ≠ > < 3
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3. จริงหรอื ไม B ถ2า > แลว2 − < −
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4. จรงิ หรือไมB ถ2า < 0 และ < 0 แล2ว > 0
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5. จรงิ หรือไมB ถ2า > 0 และ < 0 แล2ว < 0
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6. จรงิ หรือไมB 0 %
> แล2ว 2 >
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7. กรณีใดบ2าง ถ2า > แล2ว 2% < 3% เมอื่ ≠ 0 และ ≠ 0
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8 . กรณใี ดบา2 ง ถา2 > แลว2 2% > 3% เมื่อ ≠ 0 และ ≠ 0
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพมิ่ เตมิ ม.4 เทอม 1 เร่ือง จำนวนจรงิ 25
แบบฝ<กหดั 3.9ก
1. จงเขยี นชวB งตอB ไปน้ีในรูปของเซตแบบบอกเง่ือนไข พร2อมทง้ั แสดงกราฟของชวB งบนเส2นจำนวน
1) [−3, 1) 2) (−2, ∞)
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3) [4, 7] 4…………)…… ……(……−……3……,……0……)…… …………………………………………………………………………........................................................…………..
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5) (−∞, −3) 6) [1, ∞)
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7) (−1, 4] 8) (−∞, 1]
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9) (−10, −8) 10) [2.5, 4]
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2. ถ2า = (−1, 2) และ = [0, 4] จงเขยี นเซตตอB ไปน้ใี นรูปชวB ง
1) ∪ 2) ∩
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3) − 4) −
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5) ′ 6) ′
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพม่ิ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจริง 26
3. ถ2า = { | 2 ≤ ≤ 5} , = { | − 1 < < 3} และ = { | 1 < ≤ 4} จงหาเซต
ตอB ไปน ี้
1) ∪ 2) ∩
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3) ∪ ∪ 4) ∩ ∩
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5) ′ ∩ 6) ′ ∩
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7) ( ∪ ) ∩ 8) ( ∩ ) ∪ ( ∩ )
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพ่มิ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 27
แบบฝ<กหดั 3.9ข
จงหาเซตคำตอบของอสมการตBอไปน ้ี
1. 3 + 1 < 2 − 1 2. 4 + 7 > 2( + 1)
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3. 2(3 − 1) > 3( − 1) 4. 4 − (3 − ) < 3 − (3 − 2 )
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5. ! − − 6 ≤ 0 6. 2 ! + 7 + 3 ≥ 0
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7. 6 − ! ≥ 5 8. 2 < 3 − !
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เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพ่มิ เตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจรงิ 28
9. ! + 2 < 15 10. 3 ! + 2 ≥ 7
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………………………………………………………………............................……. 12. & − ! − + 1 ≥ 0
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11. & − 3 ! ≤ 10
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13. & − > 2 ! − 2
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพม่ิ เตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 29
15. (- '- % ')( !-,&) ≤ 0 16. (-,!!-) ('- &' () > 0
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17. -",-%! > 7 ","
18. - - ≤ 5
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" 20. !-- ' ' % # < 1
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เอกสารประกอบการสอนวชิ าคณิตศาสตร2เพ่ิมเตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 30
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เอกสารประกอบการสอนวชิ าคณิตศาสตร2เพิม่ เติม ม.4 เทอม 1 เร่อื ง จำนวนจริง 31
29. 4- ",&- ' %/54-",- ' "5 ≥ 0 30. ( − 1)&( + 2)# > 0
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เอกสารประกอบการสอนวิชาคณิตศาสตรเ2 พมิ่ เตมิ ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 32
แบบฝก< หดั 3.10
2) −|25| + | − 25|
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1. จงหาคาB ของ
1 ) | − 12 + 8|
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4) −|6|!
3) | − 5(10)|
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6) || − 2.5| − |3||
5) _− !"+_
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2. ให2 และ เป7นจำนวนจริง จงพจิ ารณาขอ2 ตBอไปน้ีวาB เป7นจรงิ หรอื เท็จ ถา2 เปน7 เทจ็ ให2ยกตัวอยาB งคา2 น
1) | + (− )| = | | + | − |
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3) _− 0
23_ = |'2| เมอ่ื ≠
|3|
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4) |(− )( )| = | − || |
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพ่มิ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจริง 33
5) | − − | ≤ | − | − | |
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6) ถ2า < 0 แลว2 | | <
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3. จงหาเง่อื นไขของจำนวนจรงิ และ ที่ทำให 2
1) | + | < | | + | |
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2) | + | = | | + | |
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พม่ิ เติม ม.4 เทอม 1 เร่อื ง จำนวนจริง 34
แบบฝก< หัด 3.11ก
จงหาเซตคำตอบของสมการตBอไปน ้ี
1. |2 + 1| = 3
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2. |2 − 5| = + 2
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3. |3 − 2| = − 1
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4. | | = + 2
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พม่ิ เติม ม.4 เทอม 1 เรื่อง จำนวนจรงิ 35
5. | | = 3 − 2
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6. | ! − − 4| = 2
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7. | − 1| = |2 + 1|
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8. 2| + 3| = | − 2|
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพมิ่ เติม ม.4 เทอม 1 เร่อื ง จำนวนจริง 36
แบบฝ<กหัด 3.11ข
1. จงหาเซตคำตอบของอสมการตอB ไปน ้ี
1) | − 2| < 1 2) | + 3| > 5
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3) |3 + 5| ≥ 4 4) |2 − 1| ≤ 11
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5) 2| − 2| > 6) |3 + 4| ≤ + 2
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7) |2 + 1| < 3 + 2 8) | + 1| > − 3
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพมิ่ เติม ม.4 เทอม 1 เรือ่ ง จำนวนจรงิ 37
9) | | ≥ | − 1| 10) 2| + 2| < | + 3|
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………………………………………………………………............................……. 12) 2|2 − 1| > 3| + 1|
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11) 3| − 2| ≤ | + 6|
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13) _-,-#_ > 2
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพ่มิ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 38
15) _-- ', %%_ < 1 16) _- '- !_ > 2
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2. จากการศึกษาของนกั ดาราศาสตรRพบวาB อณุ หภูมบิ นพืน้ ผิวดาวอังคารเปน7 ไปตามอสมการ | + 84| ≤ 56
เม่ือ แทนอุณหภูมิบนพ้ืนผวิ ดาวอังคาร มหี นวB ยเปน7 องศาเซลเซียส
จงหาชBวงของอณุ หภูมิบนพนื้ ผวิ ของดาวอังคารทเี่ ป7นไปไดจ2 ากอสมการน ี้
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3. ในการทดสอบวาB เหรียญเท่ยี งตรงหรือไมB ทำได2โดยทดลองโยนเหรยี ญ 100 ครั้ง แล2วบันทึก
จำนวนครั้งท่เี หรยี ญขึน้ หัว กำหนดให 2 เปน7 จำนวนคร้ังท่ีเหรยี ญขนึ้ หัว โดยทฤษฎีทางสถติ ิ กลBาววาB
ถ2าเหรยี ญท่ีนำมาทดลองโยนนนั้ ไมเB ท่ียงตรง จะได2 _- '( (/_ ≥ 1.645 จงหา ท่ที ำให2เหรียญไมBเท่ียงตรง
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพมิ่ เตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจริง 39
แบบฝก< หดั ท7ายบท
1. จงพิจารณาวาB ข2อความตอB ไปนี้เป7นจรงิ สำหรบั จำนวนจริง และ หรือไมB ถา2 เป7นเทจ็
ใหย2 กตัวอยาB งค2านประกอบ
1) ถ2า < 1 แลว2 '% > 1
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2) ถา2 ! < ! แลว2 <
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3) ถ2า > 1 และ < 1 แลว2 > 1
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2. ถ2า ! + 4 + 5 = ( + )! + ! เม่ือ > 0 จงหาคาB ของ และ
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3. จงหาผลหารและเศษเหลอื จากการหารพหุนาม ( ) ด2วยพหุนาม ( ) เมือ่ กำหนดให 2
1) ( ) = & − ! + 3 – 4 และ ( ) = – 1
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพิม่ เตมิ ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 40
2) ( ) = 4 & + 2 ! − + 6 และ ( ) = 2 + 1
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3) ( ) = ( + 2 & + 5 + 6 และ ( ) = ! − 2
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4) ( ) = # − 3 − 4 และ ( ) = 2 ! + 3
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เอกสารประกอบการสอนวิชาคณิตศาสตรเ2 พ่ิมเตมิ ม.4 เทอม 1 เร่ือง จำนวนจริง 41
5) ( ) = 2 ) − 2 # + 3 และ ( ) = − 1
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6) ( ) = $ − 3 # + 2 และ ( ) = # + 2
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7) ( ) = %/ − 2 + 1 และ ( ) = ! − 1
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พม่ิ เติม ม.4 เทอม 1 เร่ือง จำนวนจรงิ 42
8) ( ) = 3 − 3 %/ − ! และ ( ) = & + 1
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9) ( ) = %/ − 6 ) + 2 " − 8 & และ ( ) = " + & − 1
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เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพม่ิ เตมิ ม.4 เทอม 1 เร่ือง จำนวนจริง 43
4. จงหาเศษเหลอื จากการหารพหนุ าม ( ) ด2วยพหุนาม ( ) โดยใช2ทฤษฎีบทเศษเหลือ
1) ( ) = & − 3 + 15 และ ( ) = + 3
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2) ( ) = %( − 3 %! + 7 และ ( ) = − 1
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3) ( ) = " − # − 125 & + 25 ! + 75 และ ( ) = − 5
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4) ( ) = %// + 8 $) + ! − + 5 และ ( ) = + 2
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5) ( ) = " + ( − 2 และ ( ) = + เม่อื เปน7 จำนวนจรงิ
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2 และ ( ) %
6) ( )
= 4 & + − = − !
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เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พ่ิมเตมิ ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 44
5. ถ2าเศษเหลือจากการหารพหนุ าม ( ) ดว2 ย ! − 1 คือ 2 + 13 จงหา (1)
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6. ให2 ( ) เป7นพหนุ ามใด ๆ ถ2าเศษเหลอื จากการหารพหนุ าม ( ) ด2วย ! − 5 + 6 คอื 7 − 8
จงหา (2) − (3)
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7. จงหาคBา จากเงื่อนไขทีก่ ำหนดให 2
1) − หาร & − 3 เหลือเศษ 5
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2) − หาร & − 3 ! + & + ลงตวั
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เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พ่มิ เติม ม.4 เทอม 1 เรื่อง จำนวนจริง 45
8. ถ2า − เป7นตัวประกอบของ & − 3 ! + & + จงหาคBา
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9. จงแยกตวั ประกอบของพหุนามตอB ไปน ี้
1) & + 6 ! + 11 + 6
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2) & − 2 ! + 4 − 8
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3) & + 5 ! + 2 − 12
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เอกสารประกอบการสอนวิชาคณิตศาสตร2เพม่ิ เติม ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 46
4) # − & − 4 ! − 2 − 12
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5) # − 8 & + 24 ! − 32 + 16
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6) 4 & + 5 ! + 5 + 1
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เอกสารประกอบการสอนวชิ าคณิตศาสตร2เพ่ิมเติม ม.4 เทอม 1 เร่ือง จำนวนจรงิ 47
7) 2 & − ! + 6 − 3
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8) 4 # − 4 & − 3 ! + 2 + 1
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9) 2 # + 9 & − 12 ! − 29 + 30
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรเ2 พิม่ เตมิ ม.4 เทอม 1 เร่ือง จำนวนจรงิ 48
10) 2 # − 3 & − 9 ! + 9 − 2
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10. จงหาเซตคำตอบของสมการตอB ไปน้ี
1) ! − 2 − 4 = 0
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2) & − 13 + 12 = 0
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3) & + 5 ! − 2 − 24 = 0
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เอกสารประกอบการสอนวชิ าคณติ ศาสตรเ2 พ่มิ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจริง 49
4) & + 2 ! = 4 + 8
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5) & − ! − 8 + 12 = 0
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6) & − 2 + 1 = 0
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7) & = ! + + 2
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เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพ่ิมเติม ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 50
8) 4 & − 4 ! − 7 − 2 = 0
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9) 6 & + 6 = 11 ! + 1
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10) 2 # − 7 & + 9 ! − 7 + 2 = 0
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11) 4 # + 8 & + ! = 3 + 1
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12) 7 & = 2 # + 4 + 1
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