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เอกนาม

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1.1 เอกนาม

ตัวเลข  เป็นสัญลักษณ์ที่ใช้เขียนแทนจำนวน เช่น  เขียนเลข  9    แทนจำนวนสิ่งของ  เก้า  ชิ้น/อย่าง/ตัว แต่บางครั้งเราไม่สามารถใช้ตัวเลขเขียนแทนจำนวนได้ทั้งหมด  เช่น   “ห้าเท่าของจำนวนจำนวนหนึ่ง”  ไม่สามารถใช้ตัวเลขเขียนแทนจำนวนจำนวนนั้นได้  เราจะใช้ตัวอักษรภาษาอังกฤษ เช่น  a, b, c, … ,x, y, z   ตัวใดตัวหนึ่งเป็นสัญลักษณ์ที่เขียนแทนจำนวนจำนวนหนึ่ง นั่นคือใช้  5a หรือ  5b หรือ  5c หรือ  5d … หรือ  5x หรือ  5y หรือ  5z   แทน   “ห้าเท่าของจำนวนจำนวนหนึ่ง”

ข้อความที่เขียนอยู่ในรูปสัญลักษณ์ข้างต้นจะประกอบด้วย ตัวเลขและตัวอักษร เราจะเรียกตัวเลขที่ใช้เขียนแทนจำนวนว่า  ค่าคงตัว ตัวอักษร  ที่ใช้เขียนแทนจำนวนว่า  ตัวแปร และข้อความในรูปสัญลักษณ์ เช่น 3 , 5x , 7+2x  ว่านิพจน์

หลักการเขียนผลคูณระหว่างค่าคงตัวกับตัวแปร  มีดังนี้

1. กรณีที่มีค่าคงตัวมากกว่าหนึ่งตัว  ให้หาผลคูณของค่าคงตัวทั้งหมดก่อน  แล้วจึงเขียนในรูปผลคูณระหว่างค่าคงตัวกับตัวแปร  และเขียนค่าคงตัวไว้หน้าตัวแปร  
2. กรณีที่มีตัวแปรหลายตัว  ให้เขียนเรียงตามลำดับตัวอักษร  
3. กรณีที่ค่าคงตัวเป็น 1  ไม่ต้องเขียน 1 หน้าตัวแปร   ถ้าค่าคงตัวเป็น -1 ให้เขียนเฉพาะเครื่องหมายลบไว้หน้าตัวแปรทั้งหมด เช่น  1  x  y   เขียนเป็น  xy(-1) y z  x  เขียนเป็น -xyz
ข้อความในรูปประโยคสัญลักษณ์ ดังกล่าวนี้ เรียกว่า เอกนาม


สรุปว่า เอกนาม คือ นิพจน์ที่เขียนให้อยู่ในรูปการคูณกันของค่าคงตัวกับตัวแปรตั้งแต่หนึ่งตัวขึ้นไป  และเลขชี้กำลังของตัวแปรแต่ละตัวเป็นศูนย์หรือจำนวนเต็มบวกเท่านั้น เอกนามจะมีสองส่วน  คือ  1.  ส่วนที่เป็นค่าคงตัว หรือสัมประสิทธิ์ของเอกนาม 2.  ส่วนที่อยู่ ในรูปการคูณของตัวแปร โดยที่เลขชี้กำลังของตัวแปรแต่ละตัวเป็นศูนย์หรือจำนวนเต็มบวก
หมายเหตุ   หากเลขชี้กำลังของตัวแปรตัวใดตัวหนึ่งติดลบนิพจน์นั้นจะไม่เป็นเอกนาม

ดีกรีของเอกนาม เราจะเรียกผลบวกของเลขชี้กำลังของตัวแปรแต่ละตัวในเอกนาม  ว่าดีกรีของเอกนาม ดังตารางแสดงข้างล่างนี้



สำหรับดีกรีของเอกนาม 0 ไม่สามารถบอกดีกรีที่แน่นอนได้  เพราะ  0  สามารถเขียนให้อยู่ในรูปการคูณของค่าคงตัวกับตัวแปรตั้งแต่หนึ่งตัวขึ้นไป ได้  ไม่ว่า n  เป็นศูนย์หรือจำนวนเต็มบวกใด ๆ ดังนั้นจะไม่กล่าวถึงดีกรีของเอกนามศูนย์  หรือกล่าวว่า ดีกรีของเอกนาม 0 หาไม่ได้

เอกนามที่เป็นค่าคงตัวที่ไม่ใช่ศูนย์  จะมีดีกรีของเอกนามเป็น 0    เช่น  3  มีดีกรีเป็น 0  เพราะ 3 สามารถเขียนในรูปการคูณของค่าคงตัวกับตัวแปรตั้งแต่หนึ่งตัวขึ้นไปได้  



1.2  เอกนามที่คล้าย
พิจารณาเอกนามต่อไปนี้ 3xy, 4xy  จะเห็นว่าเอกนามทั้งสองนี้ต่างกันเฉพาะสัมประสิทธิ์เท่านั้น   ส่วนที่เป็นตัวแปรเหมือนกันคือ xy เรากล่าวว่าเอกนาม  3xy  และ  4xy  เป็นเอกนามที่คล้ายกัน เพราะฉะนั้น เอกนามสองเอกนามคล้ายกัน  ก็ต่อเมื่อ  เอกนามทั้งสองมีตัวแปรชุดเดียวกัน และ  เลขชี้กำลังของตัวแปรตัวเดียวกันในแต่ละเอกนามเท่ากัน ดังตัวอย่างในตาราง ทั้ง 6 ข้อ เป็นเอกนามที่คล้ายกันทั้งสิ้น



1.3  การบวกเอกนาม  มีหลักเกณฑ์ดังนี้
1.  การหาผลบวกของเอกนามที่คล้ายกัน  = ผลบวกของสัมประสิทธิ์  
2.  การหาผลบวกของเอกนามที่ไม่คล้ายกัน เช่น xy บวกกับ 2y3 ไม่สามารถเขียนผลบวกในรูปเอกนามได้แต่เขียนผลบวกในรูปการบวกได้ดังนี้  xy + 2y3
1.4  การลบเอกนาม มีหลักเกณฑ์ดังนี้
1.  ผลลบของเอกนามที่คล้ายกัน =  ผลลบของสัมประสิทธิ์
2.  การหาผลลบของเอกนามที่ไม่คล้ายกัน  เช่น   - 4x  ลบกับ  5xy  นั้น  ไม่สามารถเขียนผลลบในรูปเอกนามได้  แต่เขียนผลลบอยู่ในรูปการลบได้ดังนี้  - 4x – 5xy





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