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Sebuah polinomial  berderajat . Polinomial dibagi  bersisa  dan dibagi  bersisa . Polinomial  adalah ....

Pertanyaan

Sebuah polinomial undefined berderajat undefined. Polinomial undefined dibagi undefined bersisa begin mathsize 14px style 4 x minus 6 end style dan dibagi begin mathsize 14px style x squared minus x minus 6 end style bersisa begin mathsize 14px style 8 x minus 10 end style. Polinomial undefined adalah ....

 

  1. begin mathsize 14px style x cubed minus 2 x squared plus 3 x minus 4 end style 

  2. begin mathsize 14px style x cubed minus 3 x squared plus 2 x minus 4 end style 

  3. begin mathsize 14px style x cubed plus 2 x squared minus 3 x minus 7 end style 

  4. begin mathsize 14px style 2 x cubed plus 2 x squared minus 8 x plus 7 end style  

  5. begin mathsize 14px style 2 x cubed plus 4 x squared minus 10 x plus 9 end style 

Pembahasan Video:

Pembahasan Soal:

Perhatikan pemfaktoran berikut ini!

begin mathsize 14px style x squared minus 3 x plus 2 equals left parenthesis x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis end style

begin mathsize 14px style x squared minus x minus 6 equals left parenthesis x minus 3 right parenthesis left parenthesis x plus 2 right parenthesis end style

Diketahui bahwa P(x) dibagi begin mathsize 14px style x squared minus 3 x plus 2 end style sisa begin mathsize 14px style left parenthesis 4 x minus 6 right parenthesis end style, artinya

begin mathsize 14px style p left parenthesis 1 right parenthesis equals 4 left parenthesis 1 right parenthesis minus 6 equals negative 2 p left parenthesis 2 right parenthesis equals 4 left parenthesis 2 right parenthesis minus 6 equals 2 end style

Kemudian, P(x) dibagi begin mathsize 14px style x squared minus x minus 6 end style sisa begin mathsize 14px style left parenthesis 8 x minus 10 right parenthesis end style, artinya

begin mathsize 14px style p left parenthesis 3 right parenthesis equals 8 left parenthesis 3 right parenthesis minus 10 equals 14 p left parenthesis negative 2 right parenthesis equals 8 left parenthesis negative 2 right parenthesis minus 10 equals negative 26 end style

Misalkan begin mathsize 14px style p left parenthesis x right parenthesis equals a x cubed plus b x squared plus c x plus d end style, maka didapat persamaan berikut ini.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell p left parenthesis 1 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row cell a plus b plus c plus d end cell equals cell negative 2 space space space space space space space space space space space space space space space horizontal ellipsis space space left parenthesis 1 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell p left parenthesis 2 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row cell 8 a plus 4 b plus 2 c plus d end cell equals cell 2 space space space space space space space space space space space space space horizontal ellipsis space left parenthesis 2 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell p left parenthesis 3 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row cell 27 a plus 9 b plus 3 c plus d end cell equals cell 14 space space space space space space space space space space horizontal ellipsis space left parenthesis 3 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell p left parenthesis negative 2 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row cell negative 8 a plus 4 b minus 2 c plus d end cell equals cell negative 26 space space horizontal ellipsis space left parenthesis 4 right parenthesis end cell end table end style  

Eliminasi d

Kurangkan masing-masing persamaan (2), (3), (4) dengan persamaan (1) sehingga diperoleh persamaan baru berikut ini.

Persamaan (2) dikurangi dengan persamaan (1), maka didapat persamaan berikut ini.

begin mathsize 14px style 7 a plus 3 b plus c equals 4 space space... space space open parentheses 5 close parentheses end style 

Persamaan (3) dikurangi dengan persamaan (1), maka didapat persamaan berikut ini.

begin mathsize 14px style 26 a plus 8 b plus 2 c equals 16 space space... space left parenthesis 6 right parenthesis end style 

Persamaan (4) dikurangi dengan persamaan (1), maka didapat persamaan berikut ini.

size 14px minus size 14px 9 size 14px a size 14px plus size 14px 3 size 14px b size 14px minus size 14px 3 size 14px c size 14px equals size 14px minus size 14px 24 size 14px space size 14px space size 14px. size 14px. size 14px. size 14px space size 14px left parenthesis size 14px 7 size 14px right parenthesis 

Eliminasi c

begin mathsize 12px style stack attributes charalign center stackalign right end attributes row 26 a plus 8 b plus 2 c equals 16 open vertical bar cross times 1 close vertical bar 26 a plus 8 b plus 2 c equals 16 none end row row 7 a plus 3 b plus c equals 4 none open vertical bar cross times 2 close vertical bar 14 a plus 6 b plus 2 c equals 8 none minus end row horizontal line row 12 a plus 2 b equals 8 none none end row end stack end style

 

begin mathsize 12px style stack attributes charalign center stackalign right end attributes row minus 9 a plus 3 b minus 3 c equals negative 24 vertical line cross times 1 vertical line minus 9 a plus 3 b minus 3 c equals negative 24 none end row row 7 a plus 3 b plus c equals 4 none vertical line cross times 3 vertical line 21 a plus 9 b plus 3 c equals 12 none plus end row horizontal line row 12 a plus 12 b equals negative 12 none end row end stack end style 

Oleh karena itu, didapat 

begin mathsize 14px style 12 a plus 2 b equals 8 space space space space space space space space space... space space open parentheses 8 close parentheses 12 a plus 12 b equals negative 12 space space... space space open parentheses 9 close parentheses end style 

Kemudian, dari persamaan (8) dan persamaan (9) didapat nilai b berikut ini.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 10 b end cell equals cell negative 20 end cell row b equals cell negative 2 end cell end table end style

Substitusi begin mathsize 14px style b equals negative 2 space end styleke persamaan (8) hingga diperoleh nilai a berikut ini.

begin mathsize 14px style 12 a plus 2 left parenthesis negative 2 right parenthesis equals 8 left right double arrow 12 a equals 12 left right double arrow a equals 1 end style

Substitusi begin mathsize 14px style a equals 1 end style dan begin mathsize 14px style b equals negative 2 space end style ke persamaan (5) diperoleh nilai c berikut ini.

begin mathsize 14px style 7 left parenthesis 1 right parenthesis plus 3 left parenthesis negative 2 right parenthesis plus c equals 4 left right double arrow 1 plus c equals 4 left right double arrow c equals 3 end style

Substitusi a, b, dan c ke (1) diperoleh nilai d berikut ini.

begin mathsize 14px style 1 plus left parenthesis negative 2 right parenthesis plus 3 plus d equals negative 2 left right double arrow 2 plus d equals negative 2 left right double arrow d equals negative 4 end style

Diperoleh begin mathsize 14px style p left parenthesis x right parenthesis equals x cubed minus 2 x squared plus 4 x minus 4 end style.

Jadi, jawaban yang paling tepat adalah A.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 12 April 2021

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