Polynomial
α‘ောα်αော်αြαါ ααားαှိ polynomial αှα့် non-polynomial αှα်αွဲαော်αြαျα်αို αေ့αာαြα့်αါαα်။
| Polynomial | Non Polynomial |
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| $x^{2}-4 x+3$ | $2 x^{2}-7 x+x^{-1}$ |
| $2 x^{5}+7 x^{4}$ | $-x^{\frac{1}{2}}+x^{-3}$ |
| $5 x^{3}+6-\displaystyle\frac{5}{7} x$ | $x^{2}-\sqrt{x}$ |
| $8$ | $\displaystyle\frac{1}{x}-\displaystyle\frac{2}{x^{4}}-\sqrt[3]{x}$ |
polynomial αိုαα်αှာ αိα်းαှα်αα ်αု၏ α‘αုα်ααုα်αော αα်αိα်းαျားαာαါαော αိα်းαα်းαα ်αု αြα ်αြောα်းαွေ့ααါαα်။
α‘ောα်αော်αြαါ polynomial αို αα်αα်αေ့αာαြα့်αြαα်။
$x^3-2x^2+3x-4$- α‘αေါα်းαှα့် α‘αုα်αα္ααာαျားαြားαှိ αိα်းαုံးαျားαို terms αုαေါ်αα်။
- αိα်းαα်းαွα်αါαα်αော $x$ αို variable αုαေါ်αα်။
- $x$ ၏ α‘αြီးαုံးαα်αိα်းαို polynomial ၏ order (αို့) degree αုαေါ်αα်။
- αိα်းαှα် $x$ αါαα်αော polynomial αိα်းαα်းαα ်αုαို $f(x), g(x), P(x)$ α ααြα့် αα်αှα်αိုα်αါαα်။
| Definition: Polynomial Expression |
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| A polynomial in $x$ is an algebraic expression consisting of terms with non-negative powers of $x$ only $$ a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\ldots+a_{2} x^{2}+a_{1} x+a_{0},$$ where $n$ is a non-negative integer, the coefficients $a_n$, $a_{n-1}$,$a_{n-2}$, ..., $a_2$, $a_1$, $a_0$ are constants and $x$ is a variable. |
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Division of Polynomials
αိα်းααα်းαျားαα်းαူ Polynomial α‘αျα်းαျα်း αေါα်း၊ αုα်၊ αြှောα်၊ α ားαိုαော α‘αြေαံαုα်αောα် αျα်αျားαို αောα်αွα်αိုα်αါαα်။ αေါα်းαြα်း၊ αုα်αြα်း၊ αြှောα်αြα်း၊ αို့αို αြα်αα်းαဲ့αြီးαော α‘αα်း αျားαွα် αα်αြား αဲ့αြီး αြα ်αာ ααုαα်αα်းα ာαွα် polynomial α‘αျα်းαျα်းα ားαြα်းαိုαာ αα်αα် αα်αြαွားαါαα်။
Polynomial Long Division
Polynomial αα ်αုαို ၎α်းα‘ောα် degree αα်αော polynomial αα ်αုαြα့် α ားαျှα် αα်α α်α‘αα်း αျားαွα် αိα်းααα်းαျား α ာα‘ိα်αွဲ့၍ α ားααဲ့αို့ α ားαိုα်αα်။ αိုαဲ့αို့ α ာα‘ိα်αွဲ့၍ α ားαြα်းαို polynomial long division αုαေါ်αα်။ polynomial αျα်း αα ားαှီ αိα်းααα်းαျား α ားα‘ိα်αွဲ့α ားαြα်းαို αြα်αα်αα်αြ αါαα်။ α ားαြα်းαိုα်αာ αေါαာααျားαို αိαှိαားααါαα်။ αα်αိα်း (dividend)၊ α ား αိα်း (divisor)၊ α ားαα် (quotient)၊ α‘αြွα်း (remainder) αို့ αြα ်αြαα်။
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αα်αα်αြီး polynomial long division αို αα်αြαါαα်။ α₯ααာ $x^3 + 2x- 7$ αို $x-2$ αြα့်α ားαα်αို αါα ို့။ α‘αြွα်းαα်αျှ ααα်αို α‘ောα်αါα‘αိုα်း α ားα‘ိα်αွဲ့α ားαြα်းαြα့် αှာαူαိုα်αါαα်။
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αေးαားαော polynomial αွα် $x^2$ term αါαα်αှုααှိααြα့် α ားα‘ိα်αွဲ့ α ားαα့်α‘αါ $x^2$ term α‘αွα် αေαာαွα်αျα်၍ αော်αα်းαောα်း၊ αြှောα်αော်αိα်း 0 αား၍αော်αα်းαောα်း αြေαှα်းαိုα်αါαα်။ αα်αα်၍ $x^5-3x^4-x^3+2x^2+3x-2$ αို $x^2+2x+1$ αြα့်α ားαြα့်αါαα်။
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| αိα်းααα်းαျားα ားαြα်းαွα် α‘αြွα်းαα် α ားαα်α‘ောα် α‘αြဲαα်αြီး polynomial αျား α ားαြα်းαွα် α‘αြွα်း၏ (degree) αα် α ားαိα်း (degree) α‘ောα် α‘αြဲαα်αα်αို αွေ့ααါαα်။ |
Division Algorithm
αα်αα့်α ာαြα်းααို α‘ောα်αါαီαျှαြα်းαို α‘αြဲαြေαα်α ေαα်။
αို့αြောα့် α‘αα်αွα်αေါ်αြαဲ့αော α₯ααာαျားαို α‘ောα်αါα‘αိုα်း αီαျှαြα်းαြα့် αြα်αα်αော်αြαိုα် αါαα်။
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$a$ αα် αα်αိα်းαြα ်αြီး ၎α်းαို $b$ αြα့်α ားαα့်α‘αါ α ားαα်αှာ $q$ αြα ်αြီး α‘αြွα်းαှာ $r$ αြα ်αα် αို αါα ို့။ αိုα‘αါ $a, b, q$ αှα့် $r$ αα်αွα်αျα်αို α‘ောα်αါα‘αိုα်း αော်αြαိုα်αါαα်။
αိုαα်းαူα ွာ plynomial $f(x)$ αို $D(x)$ αြα့်α ားαα့်α‘αါ α ားαα်αှာ $Q(x)$ αြα ်αြီး α‘αြွα်းαှာ $R(x)$ αြα ်αα် αိုαါα ို့။ α‘αိုαါ function αေးαု၏ αα်αွα်αျα်αို α‘ောα်αါα‘αိုα်း αေါ်αြαိုα်αα်။
The Remainder Theorem
α‘αα်αော်αြαါ αα်αွα်αျα် αီαျှαြα်းαျားαို Division Algorithm αုαေါ်αα်။
α‘αα်၍ $D(x)=x-k$ αြα
်αျှα်
$f(x)=Q(x)(x-k)+ R$ αြα
်αα်။ remainder $R$ αα် divisor α‘ောα် degree αα်αောαြောα့် αိα်းαှα် $x$ αေါ်αွα် αှီαိုαြα်းααှိαော့αါ။ αို့αြောα့် αိα်းαေ $R$ αြα့် αα်αှα်αါαα်။ $x=k$ αြα
်αα့်α‘αါ
αို့αြောα့် polynomial $f(x)$ αို $x-k$ αြα့်α ားαα့်α‘αါ α‘αြွα်းαို $f(k)$ αုαα်αှα်αိုα်αα်။
| The Remainder Theorem |
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| If a polynomial $f (x)$ is divided by $x-k$, the remainder is $f (k)$. |
Extension of the Remainder Theorem
$f(x)$ αို $x-k$ αြα့်α ားαျှα် remainder αှာ f(k) αြα ်αြောα်းαိαှိαဲ့αြီး αြα ်αα်။ $f(x)$ αို $x+k$ αြα့်α ားαျှα် remainder αα်αို့ααα်αို αα်αα်αေ့αာαြα့်αြαα်။
| $f(x)\div (x-k)\Rightarrow \text{remainder}=f(k)$ $f(x)\div (x+k)\Rightarrow f(x)\div (x-(-k))\Rightarrow \text{remainder}=f(-k)$ |
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αα်αα်၍ $f(x)$ αို $ax-b$ αြα ်α ားαျှα် remainder αα်αို့ααα်αို αေ့αာαြα့်αြαα်။ α ားαα်αα် $Q(x)$ αြα ်αြီး α‘αြွα်းαှာ $R$ αြα ်αα် αိုαါα ို့။ αိုα‘αါ
αြα ်αα်။ αα ်αα်းαိုααော်
αြα ်αα်။ $x=\displaystyle\frac{b}{a}$ αြα ်αောα‘αါ
| $f(x)\div (ax-b)\Rightarrow \text{ remainder}=f\left( \displaystyle\frac{b}{a} \right)$ $f(x)\div (ax+b)\Rightarrow f(x)\div (ax-(-b))\Rightarrow \text{remainder}=f\left( -\displaystyle\frac{b}{a} \right)$ |
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αα်αα်၍ $f(x)$ αို $ax$ αြα ်α ားαျှα် remainder αα်αို့ααα်αို αေ့αာαြα့်αါα¦းαα်။ α ားαα်αα် $Q(x)$ αြα ်αြီး α‘αြွα်းαှာ $R$ αုαားαα်။ αို့αြောα့် $$f(x)=Q(x)(ax)+R$$ αြα ်αα်။ $x=0$ αြα ်αα့်α‘αါ
Corollary
- If a polynomial $f (x)$ is divided by $x + k$, the remainder is $f (-k)$.
- If a polynomial $f (x)$ is divided by $ax - b$, the remainder is $f\left(\displaystyle \frac{b}{a} \right)$.
- If a polynomial $f (x)$ is divided by $ax + b$, the remainder is $f\left( -\displaystyle\frac{b}{a} \right)$.
- If a polynomial $f (x)$ is divided by $ax$, the remainder is $f(0)$.
Example (1)
Using the remainder theorem, find the remainder when $x^3 + 4x^2 + 6x + 5$ is divided by $x - 2$.
Solution Let $f(x)=x^3 + 4x^2 + 6x + 5$.
When $f(x)$ is divided by $x+2$,
$\begin{aligned} \text{the remainder}&= f(2)\\\\ &=2^2+4(2)^2+6(2)+5\\\\ &=4+16+12+5\\\\ &=37 \end{aligned}$
Example (2)
If the polynomial $x^3 - 3x^2 + kx+7$ is divided by $x+3$, the remainder is $1$, find the value of $k$.
Solution Let $f(x)=x^3 - 3x^2 + kx+7$.
When $f(x)$ is divided by $x+3$, the remainder = f(-3).
By the problem,
$\begin{aligned} f(-3)&= 1\\\\ (-3)^3 - 3(-3)^2 + k(-3)+7&=1\\\\ -27-27-3k+7 &= 1\\\\ -47-3k &=1\\\\ 3k &= 48\\\\ \therefore \ \ \ k &= 16 \end{aligned}$
Example (3)
Given that $2x^4 + ax^3 - 4x^2 + bx + 7$ leaves a remainder $2x + 5$ when it is divided by $x^2 - 3x + 2$. Find the values of $a$ and $b$.
Solution
| Let $Q(x)$ be the quotient when $2x^4 + ax^3 - 4x^2 + bx + 7$ is divided by $x^2 - 3x + 2$. By division algorithm, we have $2 x^{4}+a x^{3}-4 x^{2}+b x+7=Q(x)\left(x^{2}-3 x+2\right)+2 x+5$ $2 x^{4}+a x^{3}-4 x^{2}+b x+7=Q(x)(x-1)(x-2)+2 x+5$ When $x=1$, $2+a-4+b+7=Q(1)(1-1)(1-2)+2(1)+5$ $a+b+5=0+2+5$ $a+b=2-----(1)$ When $x=2$, $2(2)^{4}+a(2)^{3}-4(2)^{2}+b(2)+7=Q(2)(2-1)(2-2)+2(2)+5$ $32+8 a-16+2 b+7=0+4+5$ $8 a+2 b+23=9$ $4 a+b=-7-----(2)$ Subtracting equation $(1)$ from equation $(2)$, $3 b=-9$ $b=-3$ Substituting $b=-3$ in equation $(1)$ $a-3=2$ $a=5$ |
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Exercise
- Using the remainder theorem, find the remainder when:
(a) $2x^2 - 13x + 10$ is divided by $x - 3$. (b) $x^3 - 3x^2 + 5x - 9$ is divided by $x - 2$. (c) $x^3 + 4x^2 + 6x + 5$ is divided by $x + 2$. (d) $x^6 - x3 - 1$ is divided by $x + 2$. (e) $9x^2 + 6x - 10$ is divided by $3x + 1$. (f) $3x^3 + 3x^2 - 11x + 8$ is divided by $3x - 1$. (g) $6x^3 + x^2 + 1$ is divided by $2x - 3$. (h) $3 (x + 4)^2 - (1 - x)^3$ is divided by $x$. (i) $(2x - 1)^3 + 6 (3 + 4x)^2 - 10$ is divided by $2x + 1$. - Find the value of $k$ if $5 x^{5}+2 k x^{3}-6 k x^{2}+9$ has a remainder of 22 when divided by $x-1$.
- The polynomial $x^{3}+a x^{2}+b x-3$ leaves a remainder of 27 when divided by $x-2$ and a remainder of 3 when divided by $x+1$. Calculate the remainder when the polynomial is divided by $x-1$.
- The expression $6 x^{2}-2 x+3$ leaves the remainder of 3 when divided by $x-p$. Determine the values of $p$.
- Given that the expression $x^{3}-a x^{2}+b x+c$ leaves the same remainder when divided by $x+1$ or $x-2$, find $a$ in terms of $b$.
- The expressions $x^{3}-7 x+6$ and $x^{3}-x^{2}-4 x+24$ have the same remainder when divided by $x+p .$ Find the possible values of $p .$
- If the polynomial $x^{3}-3 x^{2}+a x-b$ is divided by $(x-2)$ and $(x+2)$, the remainders are 21 and 1 respectively. Find the values of $a$ and $b$.
- Given that the remainder when $x^{3}-x^{2}+a x$ is divided by $x+a$ where $a>0$, is twice the remainder when it is divided by $x-2 a$, find the value of $a$.
- The remainder when $a x^{3}+b x^{2}+2 x+3$ is divided by $x-1$ is twice when it is divided by $x+1$, show that $b=3 a+3$.
- The remainder when $x^{4}+3 x^{2}-2 x+2$ is divided by $x+a$ is the square of the remainder when $x^{2}-3$ is divided by $x+a$. Calculate the possible values of $a$.
- The expression $a x^{3}-x^{2}+b x-1$ leaves the remainders of $-33$ and 77 when divided by $x+2$ and $x-3$ respectively. Find the value of $a$ and $b$ and find the remainder when the expression is divided by $x-2$.
- When the polynomial $x^{3}-3 x^{2}+k x+7$ is divided by $x+3$, the remainder is $1$ . Find the value of $k$.
- When $x^{3}+a x^{2}+b x-1$ is divided by $x-1$ the remainder is 3 and when divided by $x+2$ the remainder is $-27$. Find $a$ and $b$.
- Find the value of $\mathrm{n}$ for which the division of $x^{2 n}-7 x^{n}+5$ by $x-2$ gives the remainder of 13.
- The remainder when $a(a-b)(a+b)$ is divided by $a-2 b$ is $\displaystyle\frac{3}{4}$. Find the numerical value of $b$.
- The remainder when $p x^{3}+q x^{2}+2 x+1$ is divided by $x+1$ is twice the remainder when it is divided by $x-1$, find the relation between $p$ and $q$.
- If $f(x)=a x^{2}+b x+c$ leaves the remainders $1,25,1$ on division by $x-1$, $x+1, x-2$ respectively, show that $f(x)$ is a perfect square.
- Given that the expression $x^{2}-10 x+14$ leaves the same remainder when divided $x+2 b$ or $x+2 c$, where $b \neq c$, show that $b+c+5=0$.
- The expression $5 x^{2}-10 x+4$ has the same remainder when divided by $x-2 p$ or $x+q$ where $2 p \neq-q$. Find the value of $2 p-q$.
- When the expression $a x^{3}+5 x^{2}+b x+4$ and $b x^{3}+9 x^{2}+a x-6$ are divided by $x+3$, the remainders are $-14$ and $-12$ respectively. Find the value $a$ and $b$.
- Given that $f(x)=k x^{3}+(3 k-2) x^{2}-4$, where $k$ is a constant. If $f(x)$ is divisible by $x+2$, find the value of $k$. With this value $k$, find the remainder when $f(x)$ is divided by $(2 x-1)$.
- The expression $x^{3}+8 x^{2}+p x-25$ leaves a remainder of $R$ when divided by $x-1$ and a remainder of $-R$ when divided by $x+2$. Find the value of $p$. Hence find the remainder when the expression is divided by $x+3$.
- The polynomial $a x^{2}+b x+c$ leaves remainder 1,2, and 9 when divided by $x$, $x-1$ and $x-2$ respectively. What are the values of $a, b$ and $c$?
- If the polynomial $x^{4}-6 x^{3}+16 x^{2}-25 x+10$ is divided by another polynomial $x^{2}-2 x+k$, the remainder comes out to be $x+a$, find $k$ and $a$.
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