מתמטיקה תיכונית/אלגברה תיכונית/טכניקות אלגבריות פשוטות/חילוק רבי-אבר

הסבר

ראה גם : חילוק ארוך

שימושים

1. פונקצית חזקה.

דוגמא (מתוך בגרות במתמטיקה 006, קיץ תש"ע תרגיל 3)

בניגוד לחילוק ארוך ללא פרמטרים, בו אנו בודקים כמה פעמים נכנס המספר השמאלי בימני, בחילוק ארוך עם פרמטרים אנו מחלקים את הפרמטרים עם החזקה הגדולה ביותר מימין בשמאל. למשל, בתרגיל ${\frac {2x^{4}+4x^{3}+2x^{2}-8}{x+2}}$ הפרמטר עם החזקה הגדולה ביותר מימין הוא $\ 2x^{4}$ ומשמאל $\ x$. לכן, נחלק אותם זה בזה ונקבל $\ 2x^{3}$.

${\begin{matrix}2x^{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \end{matrix}}$

עתה נכפיל את התוצאה במספר משמאל ונחסיר אותה מהמספר שבימין :
${\begin{matrix}2x^{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \\{\underline {\ -2x^{4}+4x^{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ =\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{matrix}}$

${\begin{matrix}2x^{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \\{\underline {\ -2x^{4}+4x^{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ =\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,2x^{2}-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{matrix}}$

עתה נחזור על הפעולות; נחלק שוב את החזקה הגבוה ביותר מימין ($2x^{2}$)בחזקה הגבוה ביותר משמאל ($x$).נקבל שעלינו לחלק ב-$2x$
${\begin{matrix}2x^{3}+2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \\{\underline {\ -2x^{4}+4x^{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ =\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,2x^{2}-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{matrix}}$

נכפיל את התוצאה ונחסירה מהמספר מימין :
${\begin{matrix}2x^{3}+2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \\{\underline {\ -2x^{4}-4x^{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ =\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,2x^{2}-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\underline {-2x^{2}-4x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-4x-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{matrix}}$

נחלק $\ -4x$ (החזקה הגדולה ביותר מימין ) ב-$\ x$ החזקה הגדולה ביותר משמאל ונקבל $4$:
${\begin{matrix}2x^{3}+2x-4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \\{\underline {\ -2x^{4}-4x^{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ =\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,2x^{2}-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\underline {-2x^{2}-4x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-4x-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{matrix}}$

נכפיל את התוצאה $\ -4$ בצידו השמאלי של התרגיל $\ x+2$ ונקבל :
${\begin{matrix}2x^{3}+2x-4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\underline {\ }}{\overline {2x^{4}+4x^{3}+2x^{2}-8}}{\big \rceil }{x+2}&\ \\{\underline {\ -2x^{4}-4x^{3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ =\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,2x^{2}-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\underline {-2x^{2}-4x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-4x-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\underline {-(-4x-8)}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{matrix}}$

אין שארית ולכן הפתרון הוא : $\ 2x^{3}+2x-4$ מש"ל.

במידה ויש שארית נרשום אותה וכמובן נזכור לחלק אותה במכנה (לדוגמה אם השארית הייתה שתים הינו מוספים את האיבר ${\frac {2}{x+2}}$)