Persamaan x3−3x2−x+a=0 mempunyai 2 akar yg berlawanan. Tentukan nilai a!

Pertanyaan

Persamaan begin mathsize 14px style x cubed minus 3 x squared minus x plus a equals 0 end style mempunyai begin mathsize 14px style 2 end style akar yg berlawanan. Tentukan nilai a!

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Pembahasan

Misal akar-akar polinom begin mathsize 14px style x cubed minus 3 x squared minus x plus a equals 0 end style adalah begin mathsize 14px style x subscript 1 end stylebegin mathsize 14px style x subscript 2 end style, dan begin mathsize 14px style x subscript 3 end style.

Dengan begin mathsize 14px style a equals 1 end stylebegin mathsize 14px style b equals negative 3 end stylebegin mathsize 14px style c equals negative 1 end style, dan begin mathsize 14px style d equals a end style.

Diketahui bahwa polinom tersebut mempunyai dua akar berlawanan, sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 end cell equals cell negative x subscript 2 end cell row cell x subscript 1 plus x subscript 2 end cell equals 0 end table end style

Gunakan rumus penjumlahan akar-akar.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 plus x subscript 2 plus x subscript 3 end cell equals cell negative b over a end cell row cell 0 plus x subscript 3 end cell equals cell negative fraction numerator open parentheses negative 3 close parentheses over denominator 1 end fraction end cell row cell x subscript 3 end cell equals 3 end table end style

Karena begin mathsize 14px style x subscript 3 equals 3 end style, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 times x subscript 2 plus x subscript 1 times x subscript 3 plus x subscript 2 times x subscript 3 end cell equals cell c over a end cell row cell x subscript 1 times x subscript 2 plus open parentheses x subscript 1 plus x subscript 2 close parentheses times x subscript 3 end cell equals cell fraction numerator negative 1 over denominator 1 end fraction end cell row cell x subscript 1 times x subscript 2 plus open parentheses 0 close parentheses open parentheses 3 close parentheses end cell equals cell negative 1 end cell row cell x subscript 1 times x subscript 2 plus 0 end cell equals cell negative 1 end cell row cell x subscript 1 times x subscript 2 end cell equals cell negative 1 end cell end table end style

Kemudian pada perkalian akar-akar diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 times x subscript 2 times x subscript 3 end cell equals cell d over a end cell row cell open parentheses x subscript 1 times x subscript 2 close parentheses times x subscript 3 end cell equals cell d over a end cell row cell open parentheses negative 1 close parentheses open parentheses 3 close parentheses end cell equals cell a over 1 end cell row cell negative 3 end cell equals a end table end style

Maka, nilai begin mathsize 14px style a equals negative 3 end style.

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