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Diketahui dua polinomial masing-masing dan adalah sama. Tentukan nilai k,l,danm!

Pertanyaan

Diketahui dua polinomial masing-masing begin mathsize 14px style f open parentheses x close parentheses equals 5 x cubed plus open parentheses l plus 1 close parentheses x squared plus m x plus 12 end style dan begin mathsize 14px style g open parentheses x close parentheses equals open parentheses 2 k plus 1 close parentheses x cubed plus 5 x squared plus 2 k x plus 12 end style adalah sama. Tentukan nilai k,l,danm!

Pembahasan Soal:

Diketahui dua bentuk polinomial yang nilainya sama, maka

f(x)5x3+(l+1)x2+mx+12==g(x)(2k+1)x3+5x2+2kx+12

Ingat bahwa jika ax2=bx2, maka nilai koefisien a=b. Sehingga nilai k adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row 5 equals cell 2 k plus 1 end cell row cell 2 k end cell equals 4 row k equals 2 end table end style

nilai l :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell l plus 1 end cell equals 5 row l equals 4 end table end style

nilai m :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row m equals cell 2 k end cell row m equals cell 2 times 2 end cell row m equals 4 end table end style

Dengan demikian, didapatkan nilai k=2,l=4, dan m=4.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Kumaralalita

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 12 September 2021

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Pertanyaan yang serupa

Diketahui polinomial  dan . Jika   benilai sama untuk, nilai  adalah ...

Pembahasan Soal:

Diketahui begin mathsize 14px style f open parentheses x close parentheses equals x squared minus x plus 4 comma space g open parentheses x close parentheses equals 3 x squared minus 2 x plus c end style, dan begin mathsize 14px style f open parentheses x close parentheses comma space g open parentheses x close parentheses end style bernilai sama untuk begin mathsize 14px style x equals negative 3 end style, maka:

begin mathsize 14px style space space space space space space space space space space space space space space space space f open parentheses negative 3 close parentheses equals g open parentheses negative 3 close parentheses open parentheses negative 3 close parentheses squared minus open parentheses negative 3 close parentheses plus 4 equals 3 open parentheses negative 3 close parentheses squared minus 2 open parentheses negative 3 close parentheses plus c space space space space space space space space space space space space 9 plus 3 plus 4 equals 27 plus 6 plus c space space space space space space space space space space space space space space space space space space space space space 16 equals 33 plus c space space space space space space space space space space space space space space space space 33 plus c equals 16 space space space space space space space space space space space space space space space space space space space space space space space c equals 16 minus 33 space space space space space space space space space space space space space space space space space space space space space space space c equals negative 17 end style 

Sehingga:

begin mathsize 14px style g open parentheses x close parentheses equals 3 x squared minus 2 x plus c g open parentheses x close parentheses equals 3 x squared minus 2 x minus 17 g open parentheses 4 close parentheses equals 3 open parentheses 4 close parentheses squared minus 2 open parentheses 4 close parentheses minus 17 g open parentheses 4 close parentheses equals 3 open parentheses 16 close parentheses minus 8 minus 17 g open parentheses 4 close parentheses equals 48 minus 8 minus 17 g open parentheses 4 close parentheses equals 23 end style 

Jadi, jawaban yang benar adalah E.

Roboguru

Diketahui polinomial  dan . Jika  dan  bernilai sama untuk , tentukan:  a. nilai .

Pembahasan Soal:

Diketahui size 14px p begin mathsize 14px style left parenthesis x right parenthesis end style size 14px equals size 14px 2 size 14px x to the power of size 14px 3 size 14px minus size 14px 5 size 14px x to the power of size 14px 2 size 14px minus size 14px x size 14px plus size 14px 3 size 14px comma size 14px space size 14px q begin mathsize 14px style left parenthesis x right parenthesis end style size 14px equals size 14px 4 size 14px x to the power of size 14px 2 size 14px minus size 14px 2 size 14px x size 14px plus size 14px a, dan begin mathsize 14px style p open parentheses negative 2 close parentheses equals q open parentheses negative 2 close parentheses end style, maka:

begin mathsize 14px style space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space p open parentheses negative 2 close parentheses equals q open parentheses negative 2 close parentheses 2 open parentheses negative 2 close parentheses cubed minus 5 open parentheses negative 2 close parentheses squared minus open parentheses negative 2 close parentheses plus 3 equals 4 open parentheses negative 2 close parentheses squared minus 2 open parentheses negative 2 close parentheses plus a space space space space space space 2 left parenthesis negative 8 right parenthesis minus 5 left parenthesis 4 right parenthesis minus left parenthesis negative 2 right parenthesis plus 3 equals 4 left parenthesis 4 right parenthesis minus 2 left parenthesis negative 2 right parenthesis plus a space space space space space space space space space space space space space space space space space minus 16 minus 20 plus 2 plus 3 equals 16 plus 4 plus a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus 31 equals 20 plus a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 20 plus a equals negative 31 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals negative 31 minus 20 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals negative 51 end style  

Jadi, jawaban yang benar adalah begin mathsize 14px style bold italic a bold equals bold minus bold 51 end style.

Roboguru

Diketahui polinomial  dan . Jika  dan  bernilai sama untuk , tentukan:  b. nilai .

Pembahasan Soal:

Diketahui begin mathsize 14px style p open parentheses x close parentheses equals 2 x cubed minus 5 x squared minus x plus 3 comma space q open parentheses x close parentheses equals 4 x squared minus 2 x plus a end style, dan begin mathsize 14px style p open parentheses negative 2 close parentheses equals q open parentheses negative 2 close parentheses end style, maka:

begin mathsize 14px style space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space p left parenthesis negative 2 right parenthesis equals q left parenthesis negative 2 right parenthesis 2 left parenthesis negative 2 right parenthesis cubed minus 5 left parenthesis negative 2 right parenthesis squared minus left parenthesis negative 2 right parenthesis plus 3 equals 4 left parenthesis negative 2 right parenthesis squared minus 2 left parenthesis negative 2 right parenthesis plus a space space space space space space 2 left parenthesis negative 8 right parenthesis minus 5 left parenthesis 4 right parenthesis minus left parenthesis negative 2 right parenthesis plus 3 equals 4 left parenthesis 4 right parenthesis minus 2 left parenthesis negative 2 right parenthesis plus a space space space space space space space space space space space space space space space space space minus 16 minus 20 plus 2 plus 3 equals 16 plus 4 plus a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus 31 equals 20 plus a space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 20 plus a equals negative 31 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals negative 31 minus 20 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals negative 51 end style 

Sehingga:

begin mathsize 14px style p open parentheses x close parentheses equals 2 x cubed minus 5 x squared minus x plus 3 p open parentheses 3 close parentheses equals 2 open parentheses 3 close parentheses cubed minus 5 open parentheses 3 close parentheses squared minus open parentheses 3 close parentheses plus 3 p open parentheses 3 close parentheses equals 2 open parentheses 27 close parentheses minus 5 open parentheses 9 close parentheses minus 3 plus 3 p open parentheses 3 close parentheses equals 54 minus 45 minus 3 plus 3 p open parentheses 3 close parentheses equals 9 end style dan,

begin mathsize 14px style q open parentheses x close parentheses equals 4 x squared minus 2 x plus a q open parentheses x close parentheses equals 4 x squared minus 2 x minus 51 q open parentheses negative 1 close parentheses equals 4 open parentheses negative 1 close parentheses squared minus 2 open parentheses negative 1 close parentheses minus 51 q open parentheses negative 1 close parentheses equals 4 open parentheses 1 close parentheses plus 2 minus 51 q open parentheses negative 1 close parentheses equals 4 plus 2 minus 51 q open parentheses negative 1 close parentheses equals negative 45 end style 

dari hasil yang di atas dapat kita hitung:

begin mathsize 14px style 4 p open parentheses 3 close parentheses plus q open parentheses negative 1 close parentheses equals 4 times 9 plus open parentheses negative 45 close parentheses equals 36 minus 45 equals negative 9 end style 

Jadi, jawaban yang benar adalah begin mathsize 14px style negative 9 end style.

Roboguru

Diketahui polinomial memenuhi persamaan: Nilai  adalah ...

Pembahasan Soal:

Ingat kembali aturan penjumlahan dan pengurangan pada polinomial sebagai berikut!

Operasi penjumlahan dan pengurangan polinom dilakukan dengan cara menjumlahkan atau mengurangkan koefisien suku-suku yang mempunyai variabel dengan pangkat yang sama. 

Untuk soal di atas, maka kita memerlukan perhitungan sebagai berikut:

x cubed minus 4 x squared plus 9 x minus 5 bottom enclose space space space space space space space space space x squared minus 2 x plus 3 space minus end enclose x cubed minus 5 x squared plus 11 x minus 8

Dari penyelesaian di atas, diperoleh A x cubed plus B x squared equals x cubed minus 5 x squared dan C x plus D equals 11 x minus 8

Sehingga menggunakan konsep kesamaan diperoleh A equals 1 comma space B equals negative 5 comma space C equals 11 comma space dan space D equals negative 8

Dengan demikian, nilai 3 B minus 2 C plus D adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 B minus 2 C plus D end cell equals cell 3 open parentheses negative 5 close parentheses minus 2 open parentheses 11 close parentheses plus open parentheses negative 8 close parentheses end cell row blank equals cell negative 15 minus 22 minus 8 end cell row blank equals cell negative 45 end cell end table

Jadi, nilai bold 3 bold italic B bold minus bold 2 bold italic C bold plus bold italic D adalah bold minus bold 45.

 

Roboguru

Diketahui . Tentukan nilai a, b dan c.

Pembahasan Soal:

Ingat kembali kesamaan dua suku banyak berikut.

Jika f open parentheses x close parentheses equals g open parentheses x close parentheses maka a subscript n equals b subscript n comma space a subscript n minus 1 end subscript equals b subscript n minus 1 end subscript comma space....... space a subscript 1 equals b subscript 1

Sehingga, diperoleh perhitungan berikut ini.

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus a x y plus b y squared plus x minus 3 y minus c end cell equals cell open parentheses x plus y plus 2 close parentheses open parentheses x plus b y minus c close parentheses end cell row cell x squared plus a x y plus b y squared plus x minus 3 y minus c end cell equals cell x squared plus b x y minus c x plus x y plus b y squared minus c y plus 2 x plus 2 b y minus 2 c end cell row cell x squared plus a x y plus b y squared plus x minus 3 y minus c end cell equals cell x squared plus x y open parentheses b plus 1 close parentheses plus b y squared plus x open parentheses negative c plus 2 close parentheses minus y open parentheses c minus x minus 2 b close parentheses minus c open parentheses x plus 2 close parentheses end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared end cell equals 1 row x equals cell square root of 1 end cell row x equals cell plus-or-minus 1 end cell end table

Untuk nilai a:

table attributes columnalign right center left columnspacing 0px end attributes row cell b plus 1 end cell equals a row cell b minus a end cell equals cell negative 1 end cell row cell fraction numerator negative 1 over denominator 2 end fraction minus a end cell equals cell negative 1 end cell row cell negative a end cell equals cell negative 1 half end cell row a equals cell 1 half end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 2 end cell equals 1 row x equals cell 1 minus 2 end cell row x equals cell negative 1 end cell end table

Untuk nilai c:

table attributes columnalign right center left columnspacing 0px end attributes row cell negative c plus 2 end cell equals 1 row cell negative c end cell equals cell 1 minus 2 end cell row cell negative c end cell equals cell negative 1 end cell row c equals 1 end table

Untuk nilai b:

table attributes columnalign right center left columnspacing 0px end attributes row cell c minus x minus 2 b end cell equals 3 row cell 1 minus x minus 2 b end cell equals 3 row cell negative x minus 2 b end cell equals 2 row cell negative left parenthesis negative 1 right parenthesis minus 2 b end cell equals 2 row cell 1 minus 2 b end cell equals 2 row cell negative 2 b end cell equals 1 row b equals cell negative 1 half end cell end table

Jadi, nilai a equals 1 half, b equals negative 1 half dan c = 1.

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