Contoh Soal Polinom (Bagian 1)

 $\begin{array}{l}\\ 1.& \text{Jika}g(x)=2x^3+x^2-x+1,\\ & \text{maka}g(1)=....\\ & \begin{array}{l}\\ \text{a}.-2& & \text{d}.2\\ \text{b}.-1& \text{c}.1& \text{e}.3\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}g(x)& =2x^3+x^2-x+1\\ g(1)& =2(1{)}^3+(1{)}^2-(1)+1\\ & =2+1-1+1\\ & =3\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Jika}p(y)=5y^4+2r^2{y}^3+y^2+1\text{dan}\\ & q(y)=4y^5+3r{y}^2-3y-1\\ & \text{serta}p(-1)=q(-1),\text{maka nilai}r\\ & \text{sama dengan}....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{3}{2}\text{dan}3}& & \text{d}.-{\displaystyle \frac{3}{2}}\\ \text{b}.{\displaystyle -\frac{3}{2}\text{dan}3}& \text{c}.{\displaystyle \frac{3}{2}\text{dan}-3}& \text{e}.3\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}p(-1)& =q(-1)\\ 5(-1{)}^4+2r^2(-1{)}^3+(-1{)}^2+1& =4(-1{)}^5+3r(-1{)}^2-3(-1)-1\\ 5-2r^2+1+1& =-4+3r+3-1\\ 9-3r-2r^2& =0\\ {\displaystyle \frac{(-6-2r)(-3+2r)}{2}}& =0,\text{ingat pemfaktoran}\\ (-3-r)(-3+2r)& =0\\ r=-3\vee r& ={\displaystyle \frac{3}{2}}\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Diketahui}f(x)\text{berderajat}n.\\ & \text{Jika pembaginya berbentuk}(a{x}^2+bx+c),\\ & \text{dengan}a\ne 0,\text{maka hasil baginya berderajat}....\\ & \begin{array}{l}\\ \text{a}.n-1& & \text{d}.3\\ \text{b}.n-2& \text{c}.n-3& \text{e}.2\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Suku banyak (polinom)}\\ & =\text{pembagi}\times \text{hasil bagi}+\text{sisa}\\ & x^n+...=(a{x}^2+bx+c)\times (x^{n-2}+...)+(mx+n)\end{array}\end{array}$

$\begin{array}{l}\\ 4.& \text{Hasil bagi dan sisanya jika}(6x^4-3x^2+x-1)\\ & \text{dibagi oleh}(2x-1)\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.3x^3+{\displaystyle \frac{3}{2}{x}^2-{\displaystyle \frac{3}{4}x+\frac{1}{8}}}& \text{dan}& -{\displaystyle \frac{7}{8}}\\ \text{b}.3x^3+3x^2-{\displaystyle \frac{3}{4}x+1}& \text{dan}& -7\\ \text{c}.x^3+{\displaystyle \frac{3}{2}{x}^2-3x+\frac{1}{8}}& \text{dan}& {\displaystyle \frac{7}{8}}\\ \text{d}.x^3+{\displaystyle \frac{3}{2}{x}^2-{\displaystyle \frac{3}{4}x+1}}& \text{dan}& {\displaystyle \frac{1}{8}}\\ \text{e}.3x^3+{\displaystyle \frac{3}{2}{x}^2-{\displaystyle \frac{3}{4}x-\frac{1}{8}}}& \text{dan}& -{\displaystyle \frac{7}{8}}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \begin{array}{rrrrrrr}x=\frac{1}{2}& & 6& 0& -3& 1& -1\\ & & & 3& \frac{3}{2}& -\frac{3}{4}& \frac{1}{8}& +& \\ & & 6& 3& -\frac{3}{2}& \frac{1}{4}& \boxed{{\displaystyle -\frac{7}{8}}}\end{array}\\ & \text{Selanjutnya}\\ & \{\begin{array}{ll}\text{Hasil bagi}:& {\displaystyle \frac{6x^3+3x^2-\frac{3}{2}x+\frac{1}{4}}{2}}\\ & =3x^3+{\displaystyle \frac{3}{2}{x}^2-{\displaystyle \frac{3}{4}x+{\displaystyle \frac{1}{8}}}}\\ & \\ \text{Sisa bagi}:& -{\displaystyle \frac{7}{8}}\end{array}\end{array}\end{array}$

$\begin{array}{l}\\ 5.& \text{Hasil bagi dan sisanya jika}(x^4-x^3-x^2+x-1)\\ & \text{dibagi oleh}(x-2)(x+1)\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.x^2+1& \text{dan}& 2x+1\\ \text{b}.x^2+1& \text{dan}& 2x-1\\ \text{c}.x^2-1& \text{dan}& 2x+1\\ \text{d}.x^2-1& \text{dan}& 2x-1\\ \text{e}.2x^2-1& \text{dan}& x+1\end{array}\\ \\ & \text{Jawab}:\\ & \text{Dengan cara}\mathbf{\text{Horner-Kino}}\text{diperoleh}\end{array}$

$.\{\begin{array}{ll}\text{Suku banyak}:& f(x)=x^4-x^3-x^2+x-1\\ \text{Pembagai}:& p(x)=(x-2)(x+1)=x^2-x-2\\ & :2\text{dari}-\frac{-2}{1},\text{sedang}1=-\left(\frac{-1}{1}\right)\\ \text{Hasil bagi}:& h(x)=x^2+1\\ \text{Sisa bagi}:& s(x)=2x+1\end{array}$

$\begin{array}{rl}& \text{Sehingga},\\ & x^4-x^3-x^2+x-1\\ & =(x^2-x-2)(x^2+1)+2x+1\end{array}$