اختبار الفصل

جد ناتج جمع أو طرح المقادير الجبرية الآتية:

$(\sqrt[3]{8}{x}^2{y}^2+4xy-2)+(\sqrt[3]{27}{x}^2{y}^2-6xy+3)$

$\begin{array}{rl}& 2x^2{y}^2+4xy-2+3x^2{y}^2-6xy+3\\ & =(2x^2{y}^2+3x^2{y}^2)+(4Xy-6Xy)+(-2+3)=5x^2{y}^2-2Xy+1\end{array}$

$(\sqrt{9}{\mathrm{r}}^3{\mathrm{v}}^2+12\mathrm{gh}-6)+(\sqrt{100}{\mathrm{r}}^3{\mathrm{v}}^2-2\mathrm{gh}+1)$

$\begin{array}{rl}& 3R^3{V}^2+12GH-6+10R^3{V}^2-2GH+1\\ & =(3R^4{V}^2+10R^3{V}^2)+(12GH-2GH)+(-6+1)=13R^3{V}^2+10GH-5\end{array}$

$(\frac{1}{4}{m}^4{n}^4+7hk+8)+(\frac{1}{16}{m}^4{n}^4+9hk-12)$

$\begin{array}{rl}& \frac{1}{4}{M}^4{N}^4+7HK+8+\frac{1}{16}{M}^4{N}^4+9HK-12\\ & =(\frac{1}{4}{M}^4{N}^4+\frac{1}{16}{M}^4{N}^4)+(7HK+9HK)+(8-12)=\frac{5}{16}{M}^4{N}^4+16HK-4\end{array}$

$(\sqrt{5}{h}^2{k}^3-20x+2)+(5\sqrt{5}{h}^2{k}^3+5x-3)$

$(\sqrt{5}{H}^2{K}^3+5\sqrt{5}{H}^2{K}^3)+(-20X+5X)+(2-3)=6\sqrt{5}{H}^2{K}^3-15X-1$

$(\sqrt[3]{125}{a}^2{b}^2+4b+3)-(15a^2{b}^2+3b-6)$

$\begin{array}{rl}& (5A^2{B}^2+4B+3)+(-15A^2{B}^2-3B+6)\\ & =(5A^2{B}^2-15A^2{B}^2)+(4B-3B)+(3+6)=-10A^2{B}^2+B+9\end{array}$

$(\frac{3}{12}{r}^6{v}^3-24h-6)-(\frac{3}{12}{r}^6{v}^3-8h+1)$

$\begin{array}{rl}& =(\frac{1}{4}{R}^6{V}^3-24-6)+(-\frac{1}{4}{R}^6{V}^3+8H-1)\\ & =(\frac{1}{4}{R}^6{V}^3-\frac{1}{4}{R}^6{V}^3)+(-24H+8H)+(-6-1)=-16H-7\end{array}$

جد ناتج الضرب للحدود الجبرية الآتية:

$\left(6x^{2}y\right)\left(12xy\right)$

$(6 12)\left(x^{2+1}{y}^{1+1}\right)=72x^3{y}^2$

$\left(\frac{1}{4}\mathrm{gh}\right)\left(32\mathrm{gh}\right)$

$(\frac{1}{4} 32)G^{1+1}{H}^{1+1}=8G^2{H}^2$

$\left(\sqrt{25}{a}^4{b}^2\right)\left(5a^2{b}^2\right)$

$(\sqrt{25} 5)A^{4+2}{B}^{2+2}=25A^6{B}^4$

$\left(\sqrt[3]{1000}\right)h^4{k}^3\left(10\right)hk$

$\left(10H^4{K}^3\right)\left(10HK\right)=100M^{4+1}{K}^{3+1}=100H^5{K}^4$

$\sqrt{7}\mathrm{rv}(\sqrt{7}{\mathrm{r}}^2{\mathrm{v}}^2+\mathrm{rv}+2)$

$7R^3{V}^3+7R^2{V}^2+2\sqrt{7}RV$

$\frac{1}{9}\mathrm{wz}(81{\mathrm{w}}^{-3}{\mathrm{z}}^{-3}+\mathrm{zw}+3)$

$9w^{-2}{z}^{-2}+z^2{w}^2+\frac{1}{3}wz$

$-10z^2{w}^2(\sqrt{100}zw+10)$

$-10z^2{w}^2(102w+10)=-100z^3{w}^3-100z^2{w}^2$

جد ناتج ضرب المقدارين الجبريين لكل مما يأتي:

$(2x+y)(x+y)$

$2x^3+2x^{2}y+xy+y^2$

$(\mathrm{zw}+4)(\mathrm{zw}+5)$

$z^2{w}^2+5zw+4zw+20=z^2{w}^2+9zw+20$

$(\frac{3}{5}\mathrm{gh}-3)(\frac{1}{5}\mathrm{gh}-5)$

$$\begin{gathered} \frac{1}{25}{G}^2{H}^2-GH-\frac{3}{5}GH+15 \\ =\frac{1}{25}{G}^2{H}^2-\frac{8}{5}GH+15 \end{gathered}$$

$(3x+4)(x^2+3x+1)$

$\begin{array}{rl}& (3x+4)(x^2+3x+1)=3x(x^2+3x+1)+4(x^2+3x+1)\\ & =3x^3+9x^2+3x+4x^2+12x+4\\ & =3x^3+13x^2+15x+4\end{array}$

$(9r-1)(2r^4-3r+1)$

$\begin{array}{rl}& (9R-1)(2R^4-3R+1)=9R(2R^4-3R+1)-1(2R^4-3R+1)\\ & =18R^5-27R^2+9R-2R^4+3R-1\\ & =18R^5-2R^4-27R^2+12R-1\end{array}$

$(4m^2{n}^2-n)(4m^{-2}{n}^{-2}-n+2)$

$\begin{array}{rl}& 4M^2{N}^2(4M^{-2}{N}^{-2}-N+2)-N(4M^{-2}{N}^{-2}-N+2)\\ & =16M^0{N}^0-4M^2{N}^2+8M^2{N}^2-4M^{-2}{N}^{-1}+N^2-2N\\ & =16-4M^2{N}^2+8M^2{N}^2-4M^{-2}{N}^{-1}+N^2-2N\end{array}$

جد ناتج ضرب المقدارين الجبريين باستعمال الطريقة العمودية لكل مما يأتي:

$(\frac{4}{16}{a}^2{b}^2+\frac{25}{5})(a^2{b}^2-\frac{1}{2})$

$$\begin{gathered} \begin{array}{rl}& \frac{1}{4}{A}^2{B}^2+5\\ & A^2{B}^2-\frac{1}{9}\end{array} \\ \begin{array}{r}\frac{1}{4}{A}^4{B}^4+5A^2{B}^2\\ -\frac{1}{8}{A}^2{B}^2-\frac{5}{2}\end{array} \\ \frac{1}{4}{A}^4{B}^4+\frac{39}{8}{A}^2{B}^2-\frac{5}{2} \end{gathered}$$

$(5y^2-y)(-3y^2+y+2)$

$$\begin{gathered} \begin{array}{r}-3y^2+y+2\\ 5y^2-y\end{array} \\ \begin{array}{rl}-15y^4& +5y^3+10y^2\\ & -3y^3-y^2-2y\end{array} \\ -15y^4+2y^3+9y^2-2y \end{gathered}$$

$(8{\mathrm{z}}^{-2}{\mathrm{w}}^{-2}+4)(2\mathrm{zw}+2)$

$$\begin{gathered} \begin{array}{rl}& 8z^{-2}{w}^{-2}+4\\ & 2zw+2\end{array} \\ \begin{array}{rl}& 16z^{-1}{w}^{-1}+8zw\\ & +16z^{-2}{w}^{-2}+8\end{array} \\ 16z^{-1}{w}^{-1}+8zw+16z^{-2}{w}^{-2}+8 \end{gathered}$$

جد ناتج القسمة للمقادير الجبرية الآتية:

$\frac{-35x^4}{5x^{-2}}$

$-7x^{4+2}=-7x^6$

$\frac{12y^3}{36y^2}$

$\frac{1}{3}{y}^{3-2}=\frac{1}{3}y$

$\frac{\sqrt{100}{r}^6}{\sqrt{25}{r}^{-8}}$

$\frac{10}{5}{R}^6\cdot R^8=2R^6{R}^8$

$\frac{30v^4}{6v^{-8}}$

$\frac{30}{6}{v}^4\cdot v^8=5v^{12}$

$\frac{-35h^2}{5h^{-7}}$

$\frac{-35}{5}{H}^2\cdot H^7=-7H^9$

$\frac{56m^2{n}^4-7m^2{n}^2+42}{7mn}$

$\begin{array}{rl}& \frac{5G{M}^2{N}^4}{7MN}-\frac{7M^2{N}^2}{7MN}+\frac{42}{7MN}\\ & =8M{N}^3-MN+\frac{6}{MN}\end{array}$

$\frac{-25z{w}^2+10zw-5}{5zw}$

$\begin{array}{rl}& =\frac{-25z{w}^2}{52W}+\frac{10zw}{5zw}-\frac{5}{5zw}\\ & =-5w+2-\frac{1}{zw}\end{array}$

$\frac{81-27ab-3a}{9b}$

$\begin{array}{rl}& \frac{81}{9B}-\frac{27AB}{9B}-\frac{3A}{9B}\\ & =\frac{9}{B}-3A-\frac{9}{3B}\end{array}$

حلل المقادير الجبرية باستعمال العامل المشترك الأكبر:

$14y^2+2y-24$

$2(7y^2+y-12)$

$4z^4{w}^2-16z^2{w}^2+48zw$

$4zw(z^{3}w-4zw+12)$

$100c^3{d}^3+50c^2{d}^2+25cd$

$25CD(4C^2{D}^2+2CD+1)$

حلل المقادير الجبرية باستعمال الفرق بين مربعين:

$4z^2-16$

$\begin{array}{rl}& (2z-4)(2z+4)\end{array}$

$144x^4-64$

$(12x^2-8)(12x^2+8)$

$25r^2-3$

$(5R-\sqrt{3})(5R+\sqrt{3})$

$81{\mathrm{g}}^2{\mathrm{h}}^2-36$

$(9GH-6)(9GH+6)$

حلل المقادير باستعمال العامل المشترك الأكبر ثم الفرق بين مربعين:

$18{\mathrm{v}}^2-32$

$\begin{array}{rl}& 2(9V^2-16)\\ & =2(3V-4)(3V+4)\end{array}$

$50z^2-2$

$\begin{array}{rl}& 2(25z^2-1)\\ & =2(5z-1)(5z+1)\end{array}$

$400x-4x{y}^2$

$\begin{array}{rl}& 4x(100-y^2)\\ & =4x(10-y)(10+y)\end{array}$

$81w-169w^3$

$\begin{array}{rl}& W(81-169W^2)\\ & =W(9-13W)(9+13W)\end{array}$