বহুপদ
Polynomials
Exercise 2.3
1. \(P(x) \) বহুপদটোক \(g(x) \) বহুপদটোৰে হৰণ কৰা আৰু প্ৰতিটোৰে ক্ষেত্ৰত ভাগফল আৰু ভাগশেষ নিৰ্ণয় কৰা।
(i) \( P(x) = x^3 - 3x^2 + 5x - 3 \), \( g(x) = x^2 - 2 \)
সমাধানঃ
\[
\begin{array}{r}
x - 3 \phantom{)} \\
x^2 - 2 \overline{\smash{)} x^3 - 3x^2 + 5x - 3} \\
\underline{-(x^3 \phantom{+0x^2} - 2x)} \\
-3x^2 + 7x - 3 \\
\underline{-(-3x^2 \phantom{+7x} + 6)} \\
7x - 9 \\
\end{array}
\]
ভাগফল: \( x - 3 \)
ভাগশেষ: \( 7x - 9 \)
(ii) \( P(x) = x^4 - 3x^2 + 4x + 5 \), \( g(x) = x^2 - x - 1 \)সমাধানঃ
\[
\require{enclose}
\begin{array}{r}
x^2 - x - 3 \\
x^2 - x - 1 \enclose{longdiv}{x^4 + 0x^3 - 3x^2 + 4x + 5} \\
\underline{x^4 - x^3 - x^2} \\
-x^3 - 2x^2 + 4x \\
\underline{-x^3 + x^2 + x} \\
-3x^2 + 3x + 5 \\
\underline{-3x^2 + 3x + 3} \\
2
\end{array}
\]
(iii) \( P(x) = x^4 - 5x + 6 \), \( g(x) = x^2 + 2 \)সমাধানঃ
\[
\begin{array}{rl}
& -x^2 - 2 \\
x^2 + 2 & \enclose{longdiv}{\ x^4 + 0x^3 + 0x^2 - 5x + 6} \\
& \underline{\ \ - x^4 \phantom{+0x^3} - 2x^2 } \\
& \quad 0x^3 + 2x^2 - 5x + 6 \\
& \underline{\quad - (2x^2 \phantom{+0x^3} - 4) } \\
& \quad\quad\quad\quad -5x + 10 \\
\end{array}
\]
সমাধানঃ
$$\require{enclose}
\begin{array}{r}
2x^2 + 3x + 4 \\
x^2-3 \enclose{longdiv}{2x^4 + 3x^3 - 2x^2 - 9x - 12} \\
\underline{-(2x^4 \phantom{+3x^3} - 6x^2)} \\
\phantom{2x^4} 3x^3 + 4x^2 - 9x \\
\underline{-(3x^3 \phantom{+4x^2} - 9x)} \\
\phantom{2x^4 + 3x^3} 4x^2 - 12 \\
\underline{-(4x^2 - 12)} \\
\phantom{2x^4 + 3x^3 + 4x^2} 0 \\
\end{array}$$
সমাধানঃ
\[
\begin{array}{rl}
& x^3 - 1 \\
x^3 + 1\ & \overline{\smash{\big)}\ x^6 + 0x^5 + 0x^4 + 0x^3 + 3x^2 + 0x + 10} \\
& \underline{-(x^6 \phantom{+0x^5 + 0x^4} + x^3)} \\
& \quad -x^3 + 3x^2 + 0x + 10 \\
& \underline{\quad -(-x^3 \phantom{+3x^2} - 1)} \\
& \quad\quad\quad 3x^2 + 11 \\
\end{array}
\]
3. যদি দুটা শূণ্য \( \sqrt{\dfrac{5}{3}} \) আৰু \( -\sqrt{\dfrac{5}{3}} \),
তেন্তে
\( 3x^4 + 6x^3 - 2x^2 - 10x - 5 \) ৰ বাকী আটাইবোৰ শূণ্য উলিওৱা।
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