RD Sharma solution class 8 chapter 8 Division of Algebraic Expression Exercise 8.3

Exercise 8.3

Page-8.6

Question 1:

Divide x + 2x² + 3x⁴ − x⁵ by 2x.

Answer 1:

$$\begin{gathered} \frac{x+2x^2+3x^4-x^5}{2x} \\ =\frac{x}{2x}+\frac{2x^2}{2x}+\frac{3x^4}{2x}-\frac{x^5}{2x} \\ =\frac{1}{2}+x+\frac{3}{2}{x}^3-\frac{1}{2}{x}^{\mathit{4}} \end{gathered}$$                                                                                   

Question 2:

Divide $y^4-3y^3+\frac{1}{2}{y}^2 \mathrm{by} 3y$.

Answer 2:

$$\begin{gathered} \frac{y^{\mathit{4}}\mathit{-}\mathit{3}{y}^{\mathit{3}}\mathit{+}\frac{\mathit{1}}{\mathit{2}}{y}^{\mathit{2}}}{\mathit{3}y} \\ \mathit{=}\frac{y^{\mathit{4}}}{\mathit{3}y}\mathit{-}\frac{\mathit{3}{y}^{\mathit{3}}}{\mathit{3}y}\mathit{+}\frac{\frac{\mathit{1}}{\mathit{2}}{y}^{\mathit{2}}}{\mathit{3}y} \\ \mathit{=}\frac{\mathit{1}}{\mathit{3}}{y}^{\mathit{(}\mathit{4}\mathit{-}\mathit{1}\mathit{)}}\mathit{-}{y}^{\mathit{(}\mathit{3}\mathit{-}\mathit{1}\mathit{)}}\mathit{+}\frac{\mathit{1}}{\mathit{6}}{y}^{\mathit{(}\mathit{2}\mathit{-}\mathit{1}\mathit{)}} \\ \mathit{=}\frac{\mathit{1}}{\mathit{3}}{y}^{\mathit{3}}\mathit{-}{y}^{\mathit{2}}\mathit{+}\frac{\mathit{1}}{\mathit{6}}y \end{gathered}$$

Question 3:

Divide −4a³ + 4a² + a by 2a.

Answer 3:

$$\begin{gathered} \frac{\mathit{-}\mathit{4}{a}^{\mathit{3}}\mathit{+}\mathit{4}{a}^{\mathit{2}}\mathit{+}a}{\mathit{2}a} \\ \mathit{=}\frac{\mathit{-}\mathit{4}{a}^{\mathit{3}}}{\mathit{2}a}\mathit{+}\frac{\mathit{4}{a}^{\mathit{2}}}{\mathit{2}a}\mathit{+}\frac{a}{\mathit{2}a} \\ \mathit{=}\mathit{-}\mathit{2}{a}^{\mathit{(}\mathit{3}\mathit{-}\mathit{1}\mathit{)}}\mathit{+}\mathit{2}{a}^{\mathit{(}\mathit{2}\mathit{-}\mathit{1}\mathit{)}}\mathit{+}\frac{\mathit{1}}{\mathit{2}} \\ \mathit{=}\mathit{-}\mathit{2}{a}^{\mathit{2}}\mathit{+}\mathit{2}a\mathit{+}\frac{\mathit{1}}{\mathit{2}} \end{gathered}$$

                                                                       

Question 4:

Divide $-x^6+2x^4+4x^3+2x^2 \mathrm{by} \sqrt{2}{x}^2$.

Answer 4:

$$\begin{gathered} \frac{\mathit{-}{x}^{\mathit{6}}\mathit{+}\mathit{2}{x}^{\mathit{4}}\mathit{+}\mathit{4}{x}^{\mathit{3}}\mathit{+}\mathit{2}{x}^{\mathit{2}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}} \\ \mathit{=}\frac{\mathit{-}{x}^{\mathit{6}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\mathit{+}\frac{\mathit{2}{x}^{\mathit{4}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\mathit{+}\frac{\mathit{4}{x}^{\mathit{3}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}}\mathit{+}\frac{\mathit{2}{x}^{\mathit{2}}}{\sqrt{\mathit{2}}{x}^{\mathit{2}}} \\ \mathit{=}\frac{\mathit{-}\mathit{1}}{\sqrt{\mathit{2}}}{x}^{\mathit{(}\mathit{6}\mathit{-}\mathit{2}\mathit{)}}\mathit{+}\sqrt{\mathit{2}}{x}^{\mathit{(}\mathit{4}\mathit{-}\mathit{2}\mathit{)}}\mathit{+}\mathit{2}\sqrt{\mathit{2}}{x}^{\mathit{(}\mathit{3}\mathit{-}\mathit{2}\mathit{)}}\mathit{+}\sqrt{\mathit{2}}{x}^{\mathit{(}\mathit{2}\mathit{-}\mathit{2}\mathit{)}} \\ \mathit{=}\frac{\mathit{-}\mathit{1}}{\sqrt{\mathit{2}}}{x}^{\mathit{4}}\mathit{+}\sqrt{\mathit{2}}{x}^{\mathit{2}}\mathit{+}\mathit{2}\sqrt{\mathit{2}}x\mathit{+}\sqrt{\mathit{2}} \end{gathered}$$                                            
                                            

Question 5:

Divide 5z³ − 6z² + 7z by 2z.

Answer 5:

$$\begin{gathered} \frac{\mathit{5}{z}^{\mathit{3}}\mathit{-}\mathit{6}{z}^{\mathit{2}}\mathit{+}\mathit{7}z}{\mathit{2}z} \\ \mathit{=}\frac{\mathit{5}{z}^{\mathit{3}}}{\mathit{2}z}\mathit{-}\frac{\mathit{6}{z}^{\mathit{2}}}{\mathit{2}z}\mathit{+}\frac{\mathit{7}z}{\mathit{2}z} \\ \mathit{=}\frac{\mathit{5}}{\mathit{2}}{z}^{\mathit{(}\mathit{3}\mathit{-}\mathit{1}\mathit{)}}\mathit{-}\mathit{3}{z}^{\mathit{(}\mathit{2}\mathit{-}\mathit{1}\mathit{)}}\mathit{+}\frac{\mathit{7}}{\mathit{2}} \\ \mathit{=}\frac{\mathit{5}}{\mathit{2}}{z}^{\mathit{2}}\mathit{-}\mathit{3}z\mathit{+}\frac{\mathit{7}}{\mathit{2}} \end{gathered}$$
                                                                

Question 6:

Divide $\sqrt{3} a^4+2\sqrt{3} a^3+3a^2-6a \mathrm{by} 3a$.

Answer 6:

$$\begin{gathered} \frac{\sqrt{\mathit{3}}{a}^{\mathit{4}}\mathit{+}\mathit{2}\sqrt{\mathit{3}}{a}^{\mathit{3}}\mathit{+}\mathit{3}{a}^{\mathit{2}}\mathit{-}\mathit{6}a}{\mathit{3}a} \\ \mathit{=}\frac{\sqrt{\mathit{3}}{a}^{\mathit{4}}}{\mathit{3}a}\mathit{+}\frac{\mathit{2}\sqrt{\mathit{3}}{a}^{\mathit{3}}}{\mathit{3}a}\mathit{+}\frac{\mathit{3}{a}^{\mathit{2}}}{\mathit{3}a}\mathit{-}\frac{\mathit{6}a}{\mathit{3}a} \\ \mathit{=}\frac{\mathit{1}}{\sqrt{\mathit{3}}}{a}^{\mathit{(}\mathit{4}\mathit{-}\mathit{1}\mathit{)}}\mathit{+}\frac{\mathit{2}}{\sqrt{\mathit{3}}}{a}^{\mathit{(}\mathit{3}\mathit{-}\mathit{1}\mathit{)}}\mathit{+}{a}^{\mathit{(}\mathit{2}\mathit{-}\mathit{1}\mathit{)}}\mathit{-}\mathit{2} \\ \mathit{=}\frac{\mathit{1}}{\sqrt{\mathit{3}}}{a}^{\mathit{3}}\mathit{+}\frac{\mathit{2}}{\sqrt{\mathit{3}}}{a}^{\mathit{2}}\mathit{+}a\mathit{-}\mathit{2} \end{gathered}$$