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Faktor dari  adalah ....

Faktor dari begin mathsize 14px style fraction numerator x cubed minus 1 over denominator x to the power of 5 minus 1 end fraction end style adalah ....

Jawaban:

Untuk mempermudah penyelesaian, kita faktorkan bagian pembilang terlebih dahulu.

Untuk mencari faktor dari begin mathsize 14px style x cubed minus 1 end style, kita coba pilih begin mathsize 14px style x equals 1 end style. Dengan metode Horner, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell long division with stack on the left by table row 1 0 0 cell negative 1 end cell row blank 1 1 1 end table yields table row 1 1 1 cell space 0 end cell end table pile 1 end pile end long division end cell row cell Hasil space bagi end cell equals cell x squared plus x plus 1 end cell row Sisa equals 0 end table end style 

Karena sisa pembagian bernilai nol, maka begin mathsize 14px style x minus 1 end style adalah faktor dari begin mathsize 14px style x cubed minus 1 end style. Sementara itu, begin mathsize 14px style x squared plus x plus 1 end style tidak dapat difaktorkan. Jadi, faktor dari begin mathsize 14px style x cubed minus 1 end style adalah begin mathsize 14px style x minus 1 end style dan begin mathsize 14px style x squared plus x plus 1 end style.

Selanjutnya, kita faktorkan bagian penyebut. Untuk mencari faktor dari begin mathsize 14px style x to the power of 5 minus 1 end style, kita coba pilih begin mathsize 14px style x equals 1 end style. Dengan metode Horner, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell long division with stack on the left by table row 1 0 0 0 0 cell negative 1 end cell row blank 1 1 1 1 1 end table yields table row 1 1 1 1 1 0 end table pile 1 end pile end long division end cell row cell Hasil space bagi end cell equals cell x to the power of 4 plus x cubed plus x squared plus x plus 1 end cell row Sisa equals 0 end table end style 

Karena sisa pembagian bernilai nol, maka begin mathsize 14px style x minus 1 end style adalah faktor dari begin mathsize 14px style x to the power of 5 minus 1 end style. Sementara itu, begin mathsize 14px style x to the power of 4 plus x cubed plus x squared plus x plus 1 end style tidak dapat difaktorkan. Jadi, faktor dari begin mathsize 14px style x to the power of 5 minus 1 end style adalah begin mathsize 14px style x minus 1 end style dan begin mathsize 14px style x to the power of 4 plus x cubed plus x squared plus x plus 1 end style.

Maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator x cubed minus 1 over denominator x to the power of 5 minus 1 end fraction end cell equals cell fraction numerator up diagonal strike open parentheses x minus 1 close parentheses end strike open parentheses x squared plus x plus 1 close parentheses over denominator up diagonal strike open parentheses x minus 1 close parentheses end strike open parentheses x to the power of 4 plus x cubed plus x squared plus x plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses x squared plus x plus 1 close parentheses over denominator open parentheses x to the power of 4 plus x cubed plus x squared plus x plus 1 close parentheses end fraction end cell end table end style 

Jadi, faktor dari begin mathsize 14px style fraction numerator x cubed minus 1 over denominator x to the power of 5 minus 1 end fraction end style adalah begin mathsize 14px style fraction numerator open parentheses x squared plus x plus 1 close parentheses over denominator open parentheses x to the power of 4 plus x cubed plus x squared plus x plus 1 close parentheses end fraction end style.

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