Contoh Soal 2 Polinom

$\begin{array}{l}\\ 6.& \text{Diketahui bahwa}\\ & {\displaystyle \frac{f(x)}{x-2}=h(x)+{\displaystyle \frac{3}{x-2}}}\\ & \text{dan}{\displaystyle \frac{f(x)}{x-1}=h(x)+{\displaystyle \frac{2}{x-1},}}\\ & \text{jika}{\displaystyle \frac{f(x)}{(x-2)(x-1)}=h(x)+{\displaystyle \frac{s(x)}{(x-2)(x-1)},}}\\ & \text{maka}s(x)=....\\ & \begin{array}{l}\\ \text{a}.x+1& & \text{d}.2x-1\\ \text{b}.x+2& \text{c}.2x+1& \text{e}.x-2\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& {\displaystyle \frac{f(x)}{x-2}=h(x)+{\displaystyle \frac{3}{x-2}}}\\ & \Rightarrow f(x)=(x-2).h(x)+3\Rightarrow f(2)=3\\ & {\displaystyle \frac{f(x)}{x-1}=h(x)+{\displaystyle \frac{2}{x-1}}}\\ & \Rightarrow f(x)=(x-1).h(x)+2\Rightarrow f(1)=2\\ & {\displaystyle \frac{f(x)}{(x-2)(x-1)}=h(x)+{\displaystyle \frac{s(x)}{(x-2)(x-1)}}}\\ & \text{maka}f(x)=(x-2)(x-1).h(x)+s(x)\\ & f(x)=(x-2)(x-1).h(x)+px+q\\ & f(2)=2p+q=3\\ & f(1)=p+q=2,\\ & \text{sehingga dengan }\text{eliminasi akan diperoleh}\\ p& =1\text{dan}\\ & q=1\\ & \text{Jadi},px+q=x+1\end{array}\end{array}$

$\begin{array}{l}\\ 7.& \text{Jika}{x}^4+2mx-n\text{dibagi}{x}^2-1\\ & \text{bersisa}2x-1,\text{maka nilai}m\\ & \text{dan}n\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.m=-1\text{dan}n=2\\ \text{b}.m=1\text{dan}n=-2\\ \text{c}.m=1\text{dan}n=2\\ \text{d}.m=-1\text{dan}n=-2\\ \text{e}.m=-2\text{dan}n=1\end{array}\\ \\ & \text{Jawab}:\\ & \text{dengan Horner-Kino didapatkan}\end{array}$

$.\{\begin{array}{ll}\text{Suku banyak}:& f(x)=x^4+2mx-n\\ \text{Pembagai}:& p(x)=(x-1)(x+1)=x^2-1\\ & :1\text{dari}-\frac{-1}{1},\text{sedang}0=-\left(\frac{0}{1}\right)\\ \text{Hasil bagi}:& h(x)=x^2+1\\ \text{Sisa bagi}:& s(x)=2mx+(1-n)=2x-1\end{array}$

$.\begin{array}{rl}& \text{Sehingga},\\ & \bullet 2m=2\Rightarrow m=1\\ & \bullet 1-n=-1\Rightarrow n=2\end{array}$

$\begin{array}{l}\\ 8.& \text{Jika}f(x)=x^4-k{x}^2+5\text{habis dibagi}\\ & (x-1)\text{maka}f(x)\text{juga habis dibagi oleh}....\\ & \begin{array}{l}\\ \text{a}.x+1& & \text{d}.x+5\\ \text{b}.2x+1& \text{c}.3x+1& \text{e}.2x+5\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}f(x)& =x^4-k{x}^2+5\\ f(1)& =(1{)}^4-k(1{)}^2+5\\ 0& =1-k+5\\ k& =6\\ f(x)& =x^4-6x^2+5\\ & =(x^2-1)(x^2-5)\\ & =(x-1)(x+1)(x^2-5)\end{array}\end{array}$

$\begin{array}{l}\\ 9.& \text{Jika}(m-2)\text{adalah faktor dari}2m^3+3tm+4,\\ & \text{maka nilai}t\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{10}{3}}& & \text{d}.-{\displaystyle \frac{3}{10}}\\ \\ \text{b}.{\displaystyle \frac{1}{3}}& \text{c}.{\displaystyle \frac{3}{10}}& \text{e}.-{\displaystyle \frac{10}{3}}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}f(m)& =2m^3+3tm+4\\ f(2)& =2(2{)}^3+3t(2)+4\\ 0& =16+6t+4\\ -6t& =20\\ t& =-{\displaystyle \frac{10}{3}}\end{array}\end{array}$

$\begin{array}{l}\\ 10.& \text{(KSM MA Kab/Kota 2015)Nilai terkecil}n\\ & \text{yang mengkin sehingga}n.(n+1).(n+2)\\ & \text{habis dibagi 24 adalah}....\\ & \begin{array}{c}\\ \text{a}.1\\ \text{b}.2\\ \text{c}.3\\ \text{d}.4\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}k& ={\displaystyle \frac{n.(n+1).(n+2)}{24}}\\ & ={\displaystyle \frac{n.(n+1).(n+2)}{2.(2+1).(2+2)}}\\ & \text{maka}n=2\end{array}\end{array}$