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Persamaan polinomial   mempunyai dua akar kembar. Jika akar-akar polinomial tersebut adalah bilangan bulat, nilai  adalah ....

Pertanyaan

Persamaan polinomial begin mathsize 14px style x cubed minus 5 x squared plus 8 x plus n equals 0 end style  mempunyai dua akar kembar. Jika akar-akar polinomial tersebut adalah bilangan bulat, nilai begin mathsize 14px style n end style adalah ....

  1. begin mathsize 14px style 4 end style 

  2. begin mathsize 14px style 3 space end style 

  3. begin mathsize 14px style 2 end style 

  4. begin mathsize 14px style negative 3 end style 

  5. begin mathsize 14px style negative 4 end style 

Pembahasan Soal:

Ingat : 
Persamaan begin mathsize 14px style a x cubed plus b x cubed plus c x plus d equals 0 end style dengan akar-akar begin mathsize 14px style x subscript 1 comma space x subscript 2 comma space x subscript 3 end style berlaku:

begin mathsize 14px style x subscript 1 plus x subscript 2 plus x subscript 3 equals fraction numerator negative straight b over denominator straight a end fraction x subscript 1 times x subscript 2 plus x subscript 3 times x subscript 1 plus x subscript 2 times x subscript 3 equals straight c over straight a x subscript 1 times x subscript 2 times x subscript 3 equals fraction numerator negative straight d over denominator straight a end fraction end style

 

Misalkan begin mathsize 14px style x cubed minus 5 x squared plus 8 x plus n equals 0 end style memiliki akar a,b, dan c.
Karena akar kembar a = b, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight a plus straight b plus straight c end cell equals cell fraction numerator negative left parenthesis negative 5 right parenthesis space space space over denominator 1 end fraction not stretchy left right double arrow 2 straight a plus straight c equals 5 space horizontal ellipsis horizontal ellipsis space left parenthesis 1 right parenthesis end cell row cell straight a times straight b plus straight a times straight c plus straight b times straight c end cell equals cell 8 space space space space space space space space space space space space left right double arrow space straight a squared plus 2 ac equals 8 space horizontal ellipsis horizontal ellipsis space left parenthesis 2 right parenthesis end cell row cell straight a times straight b times straight c end cell equals cell fraction numerator negative left parenthesis straight n right parenthesis over denominator 1 end fraction space space space space space space not stretchy left right double arrow space straight a ² straight c equals negative straight n space horizontal ellipsis horizontal ellipsis space left parenthesis 3 right parenthesis end cell end table end style  

Dari Persamaan (1) dan (2)
Persamaan (1) bisa kita modifikasi menjadi begin mathsize 14px style c equals 5 minus 2 a space end style
Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a squared plus 2 a c end cell equals 8 row cell a squared plus 2 a left parenthesis 5 minus 2 a right parenthesis end cell equals 8 row cell a squared plus 10 a minus 4 a squared end cell equals 8 row cell negative 3 a squared plus 10 a minus 8 end cell equals 0 row cell left parenthesis negative 3 a plus 4 right parenthesis left parenthesis a minus 2 right parenthesis end cell equals 0 row cell right enclose a equals 4 over 3 space space end enclose space space a end cell equals 2 end table end style   

Karena akar polinomial bilangan bulat, maka dipilih begin mathsize 14px style a equals 2 end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row c equals cell 5 minus 2 a end cell row blank equals cell 5 minus 2.2 end cell row blank equals 1 end table end style 

Substitusikan nilai a dan c pada persamaan (3)

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative n end cell equals cell a squared c end cell row cell negative n end cell equals cell 2 squared times 1 end cell row cell negative n end cell equals 4 row n equals cell negative 4 end cell end table end style 

Jadi, Jawaban yang tepat adalah E.

 

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 20 Mei 2021

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