# Remainder Theorem : 2016 တကၠသိုလ္၀င္တန္းေမးခြန္း

The cubic polynomial $\displaystyle f(x)$ is such that the coefficient of $\displaystyle x^3$ is $\displaystyle -1$ and the roots of the equation $\displaystyle f(x)=0$ are $\displaystyle 1$, $\displaystyle 2$ and $\displaystyle k$. Given that $\displaystyle f(x)$ has a remainder of $\displaystyle 8$ when divided by $\displaystyle x-3$, find the value of $\displaystyle k$ and the remainder when $\displaystyle f(x)$ is divided by $\displaystyle x+3$ .

$\displaystyle f(x)$ ဟာ အႀကီးဆံုးထပ္ကိန္း $\displaystyle 3$ ထပ္ပါတဲ့ $\displaystyle \text{polynomial}$ ကိန္းတန္း တစ္ခုျဖစ္တယ္။ $\displaystyle f(x)=0$ ကို ေျဖရွင္းလို႔ရတဲ့ $\displaystyle x$ တန္ဖိုးေတြက $\displaystyle 1$, $\displaystyle 2$ နဲ႔ $\displaystyle k$ ျဖစ္ၾကတယ္။ $\displaystyle x^3$ ရဲ့ ေျမႇာက္ေဖၚကိန္းက $\displaystyle -1$ ျဖစ္တယ္။

$\displaystyle f(x)$ ကို $\displaystyle x-3$ နဲ႔ စားလို႔ရတဲ့ အႂကြင္း $\displaystyle \text{(remainder)}$ က $\displaystyle 8$ ျဖစ္တယ္ဆိုရင္ $\displaystyle k$ ကို ရွာေပးပါ။

အဲဒီေနာက္ $\displaystyle f(x)$ ကို $\displaystyle x+3$ နဲ႔ စားလို႔ရတဲ့ အႂကြင္း $\displaystyle \text{(remainder)}$ ကိုလည္း ရွာေပးပါ။

ဒီအခ်က္ေတြကိုေပါင္းလိုက္ရင္

• $\displaystyle f(x)=-1(x-1)(x-2)(x-k)$ လို႔သိရပါမယ္။

• $\displaystyle f(3)= 8$ ျဖစ္တယ္လို႔ သိရမယ္။

• $\displaystyle f(3)$ မွာ ကို $\displaystyle \text{function}$ မွာ အစားသြင္းၿပီး $\displaystyle f(3)$ နဲ႔ ညီေပးလိုက္ရင္ $\displaystyle k$ တန္ဖိုးရၿပီေပါ့။

• $\displaystyle k$ ရတဲ့အခါ $\displaystyle f(x)$ ကို $\displaystyle x+3$ နဲ႔ စားလို႔ရတဲ့ အႂကြင္း $\displaystyle \text{(remainder)}$ ဆိုတာ $\displaystyle f(-3)$ ကို ရွာခိုင္းတာ ျဖစ္ပါတယ္။
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Solution

$\displaystyle \begin{array}{l}\text{By the problem,}\\\\f(x)=-1(x-1)(x-2)(x-k)\\\\f(3)=8\\\\\therefore -1(3-1)(3-2)(3-k)=8\\\\\therefore -1(2)(1)(3-k)=8\\\\\therefore 3-k=-4\Rightarrow k=7\\\\\therefore f(x)=-1(x-1)(x-2)(x-7)\\\\\text{When }f(x)\ \text{is divided by }x+3,\\\\\text{the remainder = }f(-3)\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-1(-3-1)(-3-2)(-3-7)\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-1(-4)(-5)(-10)\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =200\end{array}$