Contoh Soal Polinom (Bagian 7)

$\begin{array}{ll}\\ 31.&\textrm{Diketahui faktor-faktor polinom}\\ & x^{3}+px^{2}-3x+q=0\: \: \textrm{adalah}\\ &(x+2)\: \: \textrm{dan}\: \: (x-3)\: .\: \textrm{Jika akar-akar}\\  &\textrm{polinom tersebut adalah}\: \:  x_{1},x_{2}\: \: \textrm{dan}\: \: x_{3},\\ &\textrm{maka nilai}\: \: x_{1}+x_{2}+x_{3}=\: ....\\ &\begin{array}{lll}\\ \textrm{a}.\quad -7&&\textrm{d}.\quad \color{red}4\\ \textrm{b}.\quad  -5&\qquad&\textrm{e}.\quad 7\\ \textrm{c}.\quad  -4\\  \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Misalkan bahwa}\: \:  f(x)=x^{3}+px^{2}-3x+q\\ &\textrm{dengan}\\ &x^{3}+px^{2}-3x+q=ax^{3}+bx^{2}+cx+d\\ &\textrm{Perhatikan bahwa}:\\ &f(-2)=(-2)^{3}+p(-2)^{2}-3(-2)+q=0\\ &\: \:  \quad\quad \Leftrightarrow -8+4p+6+q=0\\ &\: \:  \quad\quad \Leftrightarrow  4p+q=\color{red}2\: \color{black}......(1)\\ &f(3)=(3)^{3}+p(3)^{2}-3(3)+q=0\\ &  \quad\quad \Leftrightarrow 27+9p-9+q=0\\ &  \quad\quad \Leftrightarrow  9p+q=\color{red}-18\: \color{black}......(2)\\ &\textrm{Selanjutnya}\\ &\begin{array}{llllll} f(-2)&=&4p+q&=&2\\ f(3)&=&9p+q&=&-18&-\\\hline &&-5p&=&20\\ &&\qquad p&=&-4\\ &&\textrm{maka}\: \: q&=&18  \end{array}\\ &\textrm{Sehingga persamaan menjadi}\\ &x^{3}-4x^{2}-3x+18=0,\\ &\textrm{maka nilai}\\ &x_{1}+x_{2}+x_{3}=-\displaystyle \frac{b}{a}=-\frac{(-4)}{1}=4    \end{aligned}   \end{array}$.

$\begin{array}{ll}\\ 32.&\textrm{Nilai}\: \: m\: \: \textrm{agar-agar persamaan}\\ & x^{3}+3x^{2}-6x+m=0\: \: \textrm{membentuk}\\ &\textrm{barisan arirmetika adalah}\: ....\\ &\begin{array}{lll}\\ \textrm{a}.\quad 8&&\textrm{d}.\quad -2\\ \textrm{b}.\quad  6&\qquad&\textrm{e}.\quad -5\\ \textrm{c}.\quad  3\\  \end{array}\\\\ &\textrm{Jawab}:-\\ &\begin{aligned}&\textrm{Misalkan bahwa}\: \:  f(x)=x^{3}+3x^{2}-6x+m\\ &\textrm{dengan}\\ &x^{3}+3x^{2}-6x+m=ax^{3}+bx^{2}+cx+d\\ &\textrm{Jika}\: \: x_{1},x_{2},x_{3}\: \: \textrm{akar-akarnya, maka}\\ &2x_{2}=x_{1}+x_{3}\: \: \textrm{karena membentuk}\\ &\textbf{barisan aritmetika}\: .\: \textrm{Selanjutnya}\\ &\bullet \: \: x_{1}+x_{2}+x_{3}=\displaystyle -\frac{b}{a}=-\frac{3}{1}=-3\\ &\Leftrightarrow 2x_{2}+x_{2}=-3\Leftrightarrow x_{2}=-1\Leftrightarrow x_{1}+x_{3}=-2\\ &\bullet \: \: x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=\displaystyle \frac{c}{a}=\frac{-6}{1}=-6\\ &\Leftrightarrow x_{1}+x_{3}+x_{2}x_{3}=-6\Leftrightarrow -2+x_{2}x_{3}=-6\\ &\Leftrightarrow x_{2}x_{3}=-4\Leftrightarrow (-1)x_{3}=-4\Leftrightarrow x_{3}=4\\ &\bullet x_{1}+x_{3}=-2\Leftrightarrow x_{1}+4=2\Leftrightarrow x_{1}=-2\\ &\bullet x_{1}x_{2}x_{3}=-\displaystyle \frac{d}{a}=-\frac{m}{1}=-m\\ &\Leftrightarrow m=-(x_{1}x_{2}x_{3})=-(-2.-1.4)=-8     \end{aligned}   \end{array}$.

$\begin{array}{ll}\\ 33.&\textrm{Jika}\: \: \alpha ,\beta ,\gamma \: \: \textrm{merupakan akar persamaan}\\ & x^{3}-3x^{2}+4x+5=0\: \: \textrm{maka nilai}\\ &\alpha^{3} +\beta^{3} +\gamma ^{3}\: \: \textrm{adalah}\: ....\\ &\begin{array}{lll}\\ \textrm{a}.\quad -30&&\textrm{d}.\quad 28\\ \textrm{b}.\quad  \color{red}-28&\qquad&\textrm{e}.\quad 30\\ \textrm{c}.\quad  -24\\  \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui bahwa}\\ &x^{3}-3x^{2}+4x+5=ax^{3}+bx^{2}+cx+d\\ &\textrm{Karena memiliki akar-akar}\: :\: \alpha ,\beta ,\gamma ,\\  &\textrm{maka}\\ &\bullet \quad\alpha ^{3}-3\alpha ^{2}+4\alpha +5=0\\ &\bullet \quad\beta ^{3}-3\beta ^{2}+4\beta +5=0\\ &\bullet \quad\gamma ^{3}-3\gamma ^{2}+4\gamma +5=0\\ &\textrm{Jika ketiganya dijumlahkan, diperoleh}\\ &\alpha^{3} +\beta^{3} +\gamma ^{3}-3(\alpha^{2} +\beta^{2} +\gamma ^{2})\\ &+4(\alpha +\beta +\gamma )+15=0\\ &\Leftrightarrow \alpha^{3} +\beta^{3} +\gamma ^{3}=3(\alpha^{2} +\beta^{2} +\gamma ^{2})\\ &-4(\alpha +\beta +\gamma )-15\\ &\Leftrightarrow \alpha^{3} +\beta^{3} +\gamma ^{3}=3(\alpha +\beta +\gamma )^{2}\\ &\qquad -6(\alpha \beta +\alpha \gamma +\beta \gamma )-4(\alpha +\beta +\gamma )-15\\ &\Leftrightarrow \alpha^{3} +\beta^{3} +\gamma ^{3}=3\left ( -\displaystyle \frac{b}{a} \right )^{2}-6\left ( \displaystyle \frac{c}{a} \right )\\ &\qquad-4\left ( -\displaystyle \frac{b}{a} \right )-15\\ &\Leftrightarrow \alpha^{3} +\beta^{3} +\gamma ^{3}=3\left ( -(-3) \right )^{2}-6(4)\\ &\qquad -4(-(-4))-15\\ &\Leftrightarrow \alpha^{3} +\beta^{3} +\gamma ^{3}=27-24-16-15=\color{red}-28    \end{aligned}    \end{array}$.

$\begin{array}{ll}\\ 34.&\textrm{Jika}\: \: \alpha ,\beta ,\gamma \: \: \textrm{merupakan akar persamaan}\\ & x^{3}-14x^{2}+px+q=0\: \: \textrm{dengan}\\ &\alpha :\beta :\gamma =1:2:4\: ,\: \textrm{maka nilai}\: \: p-q\\ &\textrm{adalah}\: ....\\ &\begin{array}{lll}\\ \textrm{a}.\quad 160&&\textrm{d}.\quad 10\\ \textrm{b}.\quad  \color{red}120&\qquad&\textrm{e}.\quad 8\\ \textrm{c}.\quad  100\\  \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui bahwa}\\ &x^{3}-14x^{2}+px+q=ax^{3}+bx^{2}+cx+d\\ &\textrm{Karena memiliki akar-akar}\: :\: \alpha ,\beta ,\gamma ,\\  &\textrm{dengan}\: \: \alpha :\beta :\gamma =1:2:4,\: \textrm{maka}\\ &\alpha =n,\: \beta =2n,\: \gamma =4n\: .\: \textrm{Perhatikan}\\ &\alpha +\beta +\gamma =-\displaystyle \frac{b}{a}\Leftrightarrow n+2n+4n=14\\ &\Leftrightarrow 7n=14\Leftrightarrow n=2\: .\: \color{red}\textrm{Selanjutnya}\\ &\bullet \quad \alpha =2\Rightarrow  (2)^{3}-14(2)^{2}+p(2)+q=0\\ &\Leftrightarrow  8-56+2p+q=0\Leftrightarrow \color{blue}2p+q=48\\ &\bullet \quad \beta  =4\Rightarrow  (4)^{3}-14(4)^{2}+p(4)+q=0\\ &\Leftrightarrow  64-224+4p+q=0\Leftrightarrow \color{blue}4p+q=160\\ &\textrm{Selanjutnya perhatikan eliminasi berikut}\\ &\begin{array}{llllrl} 4p&+&q&=&160\\ 2p&+&q&=&48&-\\\hline 2p&&&=&112\\ &&q&=&-64&,\: \textrm{maka}\\ p&-&q&=&56&+\: 64=\color{red}120   \end{array}   \end{aligned}    \end{array}$.

$\begin{array}{ll}\\ 35.&\textrm{Jika diketahui}\: \: x^{3}-x-1=8\: \: \textrm{maka}\\ & x^{4}+x^{3}-x^{2}-2x+1\: =\:  ....\\ &\begin{array}{lll}\\ \textrm{a}.\quad 0&&\textrm{d}.\quad 8x+8\\ \textrm{b}.\quad  2&\qquad&\textrm{e}.\quad \color{red}8x+10\\ \textrm{c}.\quad  8\\  \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Diketahui bahwa}\\ &x^{3}-x-1=8\: \: \textrm{saat dikali}\: \: x\\ &\textrm{masing-masing ruas, maka}\\ &x^{4}-x^{2}-x=8x\: .\: \textrm{Jika keduanya}\\ &\textrm{dijumlahkan}\\ &x^{4}+x^{3}-x^{2}-2x-1=8x+8\\ &\Leftrightarrow x^{4}+x^{3}-x^{2}-2x-1+2=8x+8+2\\ &\Leftrightarrow x^{4}+x^{3}-x^{2}-2x+1=\color{red}8x+10   \end{aligned}   \end{array}$.