Class 8 Division of Algebraic Expressions Exercise 8.3

\begin{array}{l} \text {Q1. Divide } \quad x+2 x^{2}+3 x^{4}-x^{5} \text { by } 2 x \\ \\\text {Sol. } \quad \frac{x+2 x^{2}+3 x^{4}-x^{5}}{2 x} \\ =\frac{x}{2 x}+\frac{2 x^{2}}{2 x}+\frac{3 x^{4}}{2 x}-\frac{x^{5}}{2 x} \\ =\frac{1}{2}+x+\frac{3 x^{4-1}}{2}-\frac{1 x^{5-1}}{2} \quad \quad [\because \frac {{a}^m}{a^{n}} = a^{m-n}] \\ =\frac{1}{2}+x+\frac{3 x^{3}}{2}-\frac{1 x^{4}}{2} \\ \\\text {Q2. Divide } \quad y^{4}-3 y^{3}+\frac{1}{2} y^{2} \text { by } 3y \\ \\\text {Sol. } \quad \frac{y^{4}-3 y^{3}+\frac{ y^{3}}{2}}{3 y} \\=\frac{y^{4}}{3 y}-\frac{-3 y^{3}}{3 y}+\frac{\frac{y^{2}}{3 y}}{3 y} \\ =\frac{ y^{4-1}}{3}-y^{3-1}+\frac{y^{2-1}}{6} \quad \quad [\because \frac {{a}^m}{a^{n}} = a^{m-n}] \\ =\frac{y^{3}}{3}-y^{2}+\frac{y^{1}}{6} \\ \\\text {Q3. Divide } \quad -4 a^{3}+4 a^{2}+a \text { by } 2 a \\ \\\text {Sol. } \quad \frac{-4 a^{3}+4 a^{2}+a}{2 a} \\ =\frac{-4 a^{3}}{2 a}+\frac{4 a^{2}}{2 a}+\frac{a}{2 a} \\ =-2 a^{3-1}+2 a^{2-1}+\frac{1}{2} \quad \quad [\because \frac {{a}^m}{a^{n}} = a^{m-n}] \\ =-2 a^{2}+2a+\frac{1}{2} \\ \\\text {Q4. Divide } \quad -x^{6}+2 x^{4}+4 x^{3}+2 x^{2} \text { by } \sqrt{2} x^{2} \\ \\\text {Sol. } \quad \frac{-x^{6}+2 x^{4}+4 x^{3}+2 x^{2}}{\sqrt{2} x^{2}} \\=\frac{-x^{6}}{\sqrt{2} x^{2}}+\frac{2 x^{4}}{\sqrt{2} x^{2}}+\frac{4 x^{3}}{\sqrt{2} x^{2}}+\frac{2 x^{2}}{\sqrt{2} x^{2}} \\=\frac{-x^{6-2}}{\sqrt{2}}+ 2^{1-\frac{1}{2}} x^{4-2} + 2^{2-\frac{1}{2}} x^{3-2} + 2^{1-\frac{1}{2}} x^{2-2} \quad \quad [\because \frac {{a}^m}{a^{n}} = a^{m-n}] \\=\frac{-x^{6-2}}{\sqrt{2}}+\sqrt{2} x^{4-2}+2 \sqrt{2} x^{3-2}+\sqrt{2} x^{2-2} \\=\frac{-x^{4}}{\sqrt{2}}+\sqrt{2} x^{2}+2 \sqrt{2} x^{1}+\sqrt{2} x^{0} \\=\frac{-x^{4}}{\sqrt{2}}+\sqrt{2} x^{2}+2 \sqrt{2} x+\sqrt{2} \\ \\\text {Q5. Divide } \quad z^{3}-6 z^{2}+7 z \text { by } 2z \\ \\\text {Sol. } \quad \frac{5 z^{3}-6 z^{2}+7 z}{2 z} \\ =\frac{5 z^{3}}{2 z}-\frac{6 z^{2}}{2 z}+\frac{7 z}{2 z} \\ =\frac{5 z^{3-1}}{2}-3 z^{2-1}+\frac{7}{2} \quad \quad [\because \frac {{a}^m}{a^{n}} = a^{m-n}] \\=\frac{5 z^{2}}{2}-3 z+\frac{7}{2} \\ \\\text {Q6. Divide } \quad \sqrt{3} a^{4}+2 \sqrt{3} a^{3}+3 a^{2}-6 a \text { by } 3a \\ \\\text {Sol. } \quad \frac{\sqrt{3} a^{4}+2 \sqrt{3} a^{3}+3 a^{2}-6 a}{3 a} \\=\frac{\sqrt{3} a^{4}}{3 a}+\frac{2 \sqrt{3} a^{3}}{3 a}+\frac{3 a^{2}}{3 a}-\frac{6 a}{3 a} \\=\frac{a^{4-1}}{\sqrt{3} a}+\frac{2 a^{3-1}}{\sqrt{3} a}+a^{2-1}-2 \quad \quad [\because \frac {{a}^m}{a^{n}} = a^{m-n}] \\ =\frac{a^{3}}{\sqrt{3} a}+\frac{2 a^{2}}{\sqrt{3} a} + a -2 \\\end{array}