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Diketahui polinomial f(x)=x4−x3+ax2+x+b habis dibagi (x−3) dan bersisa −12 apabila dibagi (x−2). Tentukan akar-akar persamaan polinomial f(x)=0.

Pertanyaan

Diketahui polinomial begin mathsize 14px style f left parenthesis x right parenthesis equals x to the power of 4 minus x cubed plus a x squared plus x plus b end style habis dibagi begin mathsize 14px style left parenthesis x minus 3 right parenthesis end style dan bersisa begin mathsize 14px style negative 12 space end styleapabila dibagi begin mathsize 14px style left parenthesis x minus 2 right parenthesis end style. Tentukan akar-akar persamaan polinomial begin mathsize 14px style f left parenthesis x right parenthesis equals 0 end style.

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Jawaban

akar-akar persamaan polinomialnya adalah begin mathsize 14px style left curly bracket negative 2 comma negative 1 comma 1 comma 3 right curly bracket end style.

Pembahasan

Polinomial begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals straight x to the power of 4 minus straight x cubed plus ax squared plus straight x plus straight b end style 

Polinomial begin mathsize 14px style straight f left parenthesis straight x right parenthesis end style  habis dibagi begin mathsize 14px style left parenthesis x minus 3 right parenthesis end style, berarti:

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f left parenthesis 3 right parenthesis end cell equals 0 row cell 3 to the power of 4 minus 3 cubed plus straight a times 3 squared plus 3 plus straight b end cell equals 0 row cell 57 plus 9 straight a plus straight b end cell equals 0 row cell 9 straight a plus straight b end cell equals cell negative 57 space horizontal ellipsis left parenthesis persamaan space 1 right parenthesis end cell end table end style 

Polinomial undefined dibagi begin mathsize 14px style open parentheses x minus 2 close parentheses end style bersisa begin mathsize 14px style negative 12 end style, berarti:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f left parenthesis 2 right parenthesis end cell equals cell negative 12 end cell row cell 2 to the power of 4 minus 2 cubed plus straight a times 2 squared plus 2 plus straight b end cell equals cell negative 12 end cell row cell 10 plus 4 a plus straight b end cell equals cell negative 12 end cell row cell 4 straight a plus straight b end cell equals cell negative 22 space horizontal ellipsis left parenthesis persamaan space 2 right parenthesis end cell end table end style 

Eliminasi b dari  persamaan 1 dan 2 untuk menentukan nilai a

begin mathsize 14px style stack attributes charalign center stackalign right end attributes row 9 a plus b equals negative 57 end row row 4 a plus b equals negative 22 end row horizontal line row 5 a equals negative 35 end row row a equals negative 7 end row end stack minus end style 

Substitusi nilai begin mathsize 14px style a space end stylepada salah satu persamaan untuk menentukan nilai begin mathsize 14px style b end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 9 a plus b end cell equals cell negative 57 end cell row cell 9 open parentheses negative 7 close parentheses plus b end cell equals cell negative 57 end cell row cell negative 63 plus b end cell equals cell negative 57 end cell row b equals 6 end table end style 

Sehingga persamaan polinomialnya menjadi begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals x to the power of 4 minus x cubed minus 7 x squared plus x plus 6 end style.
Karena begin mathsize 14px style left parenthesis x minus 3 right parenthesis end style habis membagi polinom tersebut artinya dengan begin mathsize 14px style x equals 3 end style dapat dicari akar lainnya menggunakan skema Horner.

begin mathsize 14px style long division with stack on the left by table row 1 row vertical ellipsis end table number space table row cell negative 1 end cell row cell space space 3 end cell end table number space table row cell negative 7 end cell row cell space 6 end cell end table number space table row 1 row cell negative 3 end cell end table number space table row 6 row cell negative 6 end cell end table yields space 1 number space space space space space space 2 number space space space minus 1 number space space minus 2 number space space space space space space 0 pile 3 end pile end long division plus end style 

Diperoleh hasil bagi begin mathsize 14px style x cubed plus 2 x squared minus x minus 2 end style, sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses x cubed plus 2 x squared minus x minus 2 close parentheses end cell equals 0 row cell x squared open parentheses x plus 2 close parentheses minus 1 open parentheses x plus 2 close parentheses end cell equals 0 row cell open parentheses x plus 2 close parentheses open parentheses x squared minus 1 close parentheses end cell equals 0 row cell open parentheses x plus 2 close parentheses open parentheses x plus 1 close parentheses open parentheses x minus 1 close parentheses end cell equals 0 row cell x equals negative 2 comma space x equals minus 1 comma space x end cell equals 1 end table end style  

Jadi akar-akar persamaan polinomialnya adalah begin mathsize 14px style left curly bracket negative 2 comma negative 1 comma 1 comma 3 right curly bracket end style.

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