جواب تمرین صفحه 129 درس 7 ریاضی نهم (عبارت های گویا)

\(\frac{{ - 2{x^2}{y^3}{z^7}}}{{18x{z^5}}} = \frac{{ - 2}}{{18}} \times \frac{{{x^2}}}{x} \times \frac{{{y^3}}}{1} \times \frac{{{z^7}}}{{{z^5}}} = \) الف

\(\frac{{ - 1}}{9} \times \frac{x}{1} \times \frac{{{y^3}}}{1} \times \frac{{{z^2}}}{1} = \frac{{ - x{y^3}{z^2}}}{9}\)

\(\frac{{2{a^3}y - {a^4}{y^2} + 15xy}}{{ - 5{y^2}}} = \) ب

\(\begin{array}{l}\frac{{2{a^3}y}}{{ - 5{y^2}}} + \frac{{ - {a^4}{y^2}}}{{ - 5{y^2}}} + \frac{{15xy}}{{ - 5{y^2}}} = \\\\\frac{{ - 2{a^3}}}{{5y}} + \frac{{{a^4}}}{5} - \frac{{3x}}{y}\end{array}\)

\(\left( {{x^2} - 27} \right) \div \left( {x - 2} \right) = \frac{{{x^2} - 27}}{{x - 2}} = \) ج

\(\begin{array}{l}\frac{{{x^2} - 4 - 23}}{{x - 2}} = \frac{{{x^2} - 4}}{{x - 2}} + \frac{{ - 23}}{{x - 2}} = \\\\\frac{{(x - 2)(x + 2)}}{{x - 2}} + \frac{{ - 23}}{{x - 2}} = \\\\x + 2 - \frac{{23}}{{x - 2}}\end{array}\)

\(\left( {3{y^2} - 10y - 24} \right) \div \left( {3y - 4} \right):\) د

\(\begin{array}{l}\,\,\,3{y^2} - 10y - 24\,\,\,\left| {\underline {\,\,\,3y - 4\,\,\,} } \right.\\\underline { - 3{y^2} + 4y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,\,\,\,y - 2\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 6y - 24\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 6y - 8\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32\\\\\left( {3{y^2} - 10y - 24} \right) \div \left( {3y - 4} \right) = \\\\\frac{{3{y^2} - 10y - 24}}{{3y - 4}} = y - 2 - \frac{{32}}{{3y - 4}}\end{array}\)

هـ

\(\begin{array}{l}\,\,\,\,\,2{x^5} + 5{x^4} - 2{x^3} + 2{x^2} - 2x + 3\;\underline {\left| {\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x + 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \right.} \;\\\underline { - 2{x^5} - 6{x^4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,2{x^4} - {x^3} + {x^2} - x + 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {x^4} - 2{x^3} + 2{x^2} - 2x + 3\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {x^4} + 3{x^3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^3} + 2{x^2} - 2x + 3\,\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {x^3} - 3{x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {x^2} - 2x + 3\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 3x\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x + 3\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - x - 3\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\)

\(\begin{array}{l} - 3{x^4} + 4{x^6} + {x^2} + 5\;\;\left| {\underline {\,\,\,\,\,\,1 - {x^3}\,\,\,} } \right.\;\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Downarrow \\\,\,\,4{x^6} - 3{x^4} + {x^2} + 5\,\,\,\,\,\;\,\,\,\,\,\,\,\;\left| {\underline {\,\,\,\,\,\,\,\, - {x^3} + 1\,\,\,\,\,\,\,\,\,} } \right.\\\underline { - 4{x^6} + 4{x^3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\, - 4{x^3} + 3x - 4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 3{x^4} + 4{x^3} + {x^2} + 5\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 3{x^4} - 3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 4{x^3} + {x^2} - 3x + 5\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 4{x^3} + 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} - 3x + 9\end{array}\)

\( = - 4{x^3} + 3x - 4\) خارج قسمت

\( = {x^2} - 3x + 9\) باقیمانده

حال درستی عمل تقسیم را با نوشتن روابط تقسیم نشان می دهیم:

\(\begin{array}{l}( - 3{x^4} + 4{x^6} + {x^2} + 5) = (1 - {x^3})( - 4{x^3} + 3x - 4) + {x^2} - 3x + 9 = \\\\ - 3{x^4} + 4{x^6} + 3x - 4 + {x^2} - 3x + 9 = - 3{x^4} + 4{x^6} + {x^2} + 5\end{array}\)

ارتفاع × عرض × طول = حجم مکعب مستطیل

(ارتفاع × طول) ÷ حجم مکعب مستطیل = عرض

\( = \frac{{2{x^3} + 15{x^2} + 28x}}{{(x + 4)x}} = \) عرض

\(\begin{array}{l} = \frac{{2{x^3} + 15{x^2} + 28x}}{{{x^2} + 4x}}:\\\\\,\,\,\,\,2{x^3} + 15{x^2} + 28x\,\,\,\,\,\,\,\left| {\underline {\,\,{x^2} + 4x\,\,} } \right.\\\underline { - 2{x^3} - 8{x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,\,\,\,2x + 7\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7{x^2} + 28x\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 7{x^2} - 28x\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\)

\( = 2x + 7\) عرض

برای حل این مسئله بایستی باقیماندۀ تقسیم چند جمله ای \(20{x^3} + 23{x^2} - 10x + a\) بر 4x+3 برابر با صفر باشد؛ بنابراین داریم:

\(\begin{array}{l}\,\,\,\,\,20{x^3} + 23{x^2} - 10x + a\,\,\,\left| {\underline {\,\,\,\,\,\,\,\,\,\,4x + 3\,\,\,\,\,\,\,\,\,\,} } \right.\\\underline { - 20{x^3} - 15{x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,5{x^2} + 2x - 4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8{x^2} - 10x + a\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 8{x^2} - 6x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 16x + a\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 16x + 12\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a + 12\\\\a + 12 = 0 \Rightarrow a = - 12\end{array}\)

خارج قسمت و باقیماندهٔ تقسیم عبارت \(2{x^2} - 9x + 9\) بر عبارت x+3 به صورت زیر می شود:

\(\begin{array}{l}\,\,\,\,\,\,2{x^2} - 9x + 9\,\,\,\,\left| {\underline {\,\,\,\,\,\,x + 3\,\,\,\,\,} } \right.\\\underline { - \,2{x^2} - 6x\,\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,\,2x - 15\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 15x + 9\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 15x + 45\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,54\end{array}\)

خارج قسمت = 2x-15

باقیمانده = 54

خارج قسمت و باقیماندهٔ تقسیم عبارت \(2{x^2} - 9x + 9\) بر عبارت x-3 به صورت زیر می شود:

\(\begin{array}{l}\,\,\,\,\,\,2{x^2} - 9x + 9\,\,\,\,\left| {\underline {\,\,\,\,x - 3\,\,\,\,} } \right.\\\underline { - \,2{x^2} + 6x\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,\,2x - 3\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 3x + 9\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 3x - 9\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\)

خارج قسمت = 2x-3

باقیمانده = 0

خارج قسمت و باقیماندهٔ تقسیم عبارت \(2{x^2} - 9x + 9\) بر عبارت 2x-3 به صورت زیر می شود:

\(\begin{array}{l}\,\,\,\,\,\,2{x^2} - 9x + 9\,\,\,\,\left| {\underline {\,\,\,\,2x - 3\,\,\,} } \right.\\\underline { - \,2{x^2} + 3x\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,\,\,\,\,x - 3\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 6x + 9\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 6x - 9\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\)

خارج قسمت = x-3

باقیمانده = 0

خارج قسمت و باقیماندهٔ تقسیم عبارت \(2{x^2} - 9x + 9\) بر عبارت 2x+3 به صورت زیر می شود:

\(\begin{array}{l}\,\,\,\,\,\,2{x^2} - 9x + 9\,\,\,\,\left| {\underline {\,\,\,\,2x + 3\,\,\,\,} } \right.\\\underline { - \,2{x^2} - 3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \,\,\,\,\,\,\,\,x - 6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 12x + 9\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 12x + 18\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,27\end{array}\)

خارج قسمت = x-6

باقیمانده = 27