Roboguru

Jika  diekspansikan dalam pangkat naik dari , maka ekspansinya adalah  Tentukan nilai-nilai .

Pertanyaan

Jika begin mathsize 14px style open parentheses 1 plus straight a x close parentheses to the power of 8 end style diekspansikan dalam pangkat naik dari begin mathsize 14px style x end style, maka ekspansinya adalah begin mathsize 14px style straight A plus straight B x plus straight C x squared plus 7 x cubed plus... end style Tentukan nilai-nilai begin mathsize 14px style straight a comma space straight A comma space straight B comma space dan space straight C end style

Pembahasan Soal:

Dengan menggunakan formula binomial Newton yaitu 

begin mathsize 14px style left parenthesis straight a plus straight b right parenthesis to the power of straight n equals sum from straight r equals 0 to straight n of straight C subscript straight r superscript straight n times straight a to the power of straight n minus straight r end exponent times straight b to the power of straight r end style  

Maka begin mathsize 14px style open parentheses 1 plus ax close parentheses to the power of 8 end style dalam ekspansi pangkat naik dari begin mathsize 14px style x end style dapat ditulis 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis 1 plus ax right parenthesis to the power of 8 end cell equals cell sum from straight r equals 0 to 8 of straight C subscript straight r superscript straight n times left parenthesis 1 right parenthesis to the power of straight n minus straight r end exponent times left parenthesis ax right parenthesis to the power of straight r space end cell row blank equals cell sum from straight r equals 0 to 8 of straight C subscript straight r superscript straight n times left parenthesis ax right parenthesis to the power of straight r space end cell end table end style  

Suku keempat dari ekspansi tersebut adalah begin mathsize 14px style 7 x cubed end styleundefined maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight C subscript 3 superscript 8 open parentheses ax close parentheses cubed end cell equals cell 7 straight x cubed end cell row cell fraction numerator 8 factorial over denominator open parentheses 8 minus 3 close parentheses factorial times 3 factorial end fraction times straight a cubed times straight x cubed end cell equals cell 7 straight x cubed end cell row cell fraction numerator 8 times 7 times 6 times 5 factorial over denominator 5 factorial times 3 times 2 times 1 end fraction times straight a cubed times straight x cubed end cell equals cell 7 straight x cubed end cell row cell 8 times 7 times straight a cubed times straight x cubed end cell equals cell 7 straight x cubed end cell row cell 8 times straight a cubed end cell equals 1 row cell straight a cubed end cell equals cell 1 over 8 end cell row straight a equals cell 1 half end cell end table end style 

Suku pertama adalah begin mathsize 14px style straight A end style, dimana begin mathsize 14px style straight r equals 0 end style, maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight C subscript 0 superscript 8 open parentheses ax close parentheses to the power of 0 end cell equals straight A row cell open parentheses 1 close parentheses times open parentheses 1 close parentheses end cell equals straight A row straight A equals 1 end table end style 

Suku kedua adalah begin mathsize 14px style text B end text x end style, dimana begin mathsize 14px style straight r equals 1 end style, maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 1 superscript 8 times open parentheses ax close parentheses to the power of 1 end cell equals cell straight B times x end cell row cell open parentheses fraction numerator 8 factorial over denominator open parentheses 8 minus 7 close parentheses times 7 factorial end fraction close parentheses times open parentheses 1 half x close parentheses end cell equals cell straight B times x end cell row cell fraction numerator 8 times 7 factorial over denominator 1 times 7 factorial end fraction times 1 half times x end cell equals cell straight B times x end cell row cell 8 over 2 times x end cell equals cell straight B times x end cell row cell 4 times x end cell equals cell straight B times x end cell row straight B equals 4 end table end style 

Suku ketiga adalah begin mathsize 14px style straight C x squared end style, dimana begin mathsize 14px style straight r equals 2 end style, maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 2 superscript 8 times open parentheses straight a x close parentheses squared end cell equals cell straight C x squared end cell row cell open parentheses fraction numerator 8 factorial over denominator open parentheses 8 minus 2 close parentheses factorial times 2 factorial end fraction close parentheses times open parentheses 1 half x close parentheses squared end cell equals cell straight C x squared end cell row cell fraction numerator 8 times 7 times 6 factorial over denominator 6 factorial times 2 times 1 end fraction times 1 fourth times x squared end cell equals cell straight C times straight x squared end cell row cell 7 x squared end cell equals cell straight C x squared end cell row straight C equals 7 end table end style 

Jadi begin mathsize 14px style straight a equals 1 half comma space straight A equals 1 comma space straight B equals 4 comma space dan space straight C equals 7 end style.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 15 Maret 2021

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