RD Sharma solution class 8 chapter 8 Division of Algebraic Expression Exercise 8.1

Exercise 8.1

Page-8.2

Question 1:

Write the degree of each of the following polynomials.
(i) 2x² + 5x² − 7
(ii) 5x² − 3x + 2
(iii) 2x + x² − 8
(iv) $\frac{1}{2}{y}^7-12y^6+48y^5-10$
(v) 3x³ + 1
(vi) 5
(vii) 20x³ + 12x²y² − 10y² + 20

Answer 1:

$$\begin{gathered} \left(\mathrm{i}\right) \mathit{C}\mathit{o}\mathit{r}\mathit{r}\mathit{e}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{ } : It is 2{\mathrm{x}}^3+5{\mathrm{x}}^2-7 instead of 2{\mathrm{x}}^2+5{\mathrm{x}}^2-7. \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polymonial} 2{\mathrm{x}}^3+5{\mathrm{x}}^2-7 \mathrm{is} 3. \\ \left(\mathrm{ii}\right) \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polymonial} 5{\mathrm{x}}^2-35\mathrm{x}+2 \mathrm{is} 2. \\ \left(\mathrm{iii}\right) \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polymonial} 2\mathrm{x}+{\mathrm{x}}^2-8 \mathrm{is} 2. \\ \left(\mathrm{iv}\right) \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polymonial} \frac{1}{2}{\mathrm{y}}^7-12{\mathrm{y}}^6+48{\mathrm{y}}^5-10 \mathrm{is} 7. \\ \left(\mathrm{v}\right) \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polymonial} 3{\mathrm{x}}^3+1 \mathrm{is} 3. \\ \left(\mathrm{vi}\right) 5 \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{polynomial} \mathrm{and} \mathrm{its} \mathrm{degree} \mathrm{is} 0. \\ \left(\mathrm{vii}\right) \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polymonial} 20{\mathrm{x}}^3+12{\mathrm{x}}^2{\mathrm{y}}^2-10{\mathrm{y}}^2+20 \mathrm{is} 4. \end{gathered}$$

Question 2:

Which of the following expressions are not polynomials?
(i) x² + 2x⁻²
(ii) $\sqrt{ax}+x^2-x^3$
(iii) 3y³ − $\sqrt{5}y$ + 9
(iv) ax^1/2 + ax + 9x² + 4
(v) 3x⁻² + 2x⁻¹ + 4x +5

Answer 2:

$$\begin{gathered} \left(\mathrm{i}\right) {\mathrm{x}}^2+2{\mathrm{x}}^{-2} \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{polynomial} \mathrm{because} -2 \mathrm{is} \mathrm{the} \mathrm{power} \mathrm{of} \mathrm{variable} \mathrm{x} \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{non} \mathrm{negative} \mathrm{integer}. \\ \left(\mathrm{ii}\right) \sqrt{\mathrm{ax}}+{\mathrm{x}}^2-{\mathrm{x}}^3 \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{polynomial} \mathrm{because} \frac{1}{2} \mathrm{is} \mathrm{the} \mathrm{power} \mathrm{of} \mathrm{variable} \mathrm{x} \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{non} \mathrm{negative} \mathrm{integer}. \\ \left(\mathrm{iii}\right) 3{\mathrm{y}}^3-\sqrt{5}\mathrm{y}+9 \mathrm{is} \mathrm{a} \mathrm{polynomial} \mathrm{because} \mathrm{the} \mathrm{powers} \mathrm{of} \mathrm{variable} \mathrm{y} \mathrm{are} \mathrm{non} \mathrm{negative} \mathrm{integer}\text{s}. \\ \left(\mathrm{iv}\right) {\mathrm{ax}}^{\frac{1}{2}}+\mathrm{ax}+9{\mathrm{x}}^2+4 \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{polynomial} \mathrm{because} \frac{1}{2} \mathrm{is} \mathrm{the} \mathrm{power} \mathrm{of} \mathrm{variable} \mathrm{x} \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{non} \mathrm{negative} \mathrm{integer}. \\ \left(\mathrm{v}\right) 3{\mathrm{x}}^{-2}+2{\mathrm{x}}^{-1}+4\mathrm{x}+5 \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{polynomial} \mathrm{because} -2 \mathrm{and} -1 \mathrm{are} \mathrm{the} \mathrm{powers} \mathrm{of} \mathrm{variable} \mathrm{x} \mathrm{are} \mathrm{not} \mathrm{non} \mathrm{negative} \mathrm{integer}s. \end{gathered}$$

Question 3:

Write each of the following polynomials in the standard form. Also, write their degree.
(i) x² + 3 + 6x + 5x⁴
(ii) a² + 4 + 5a⁶
(iii) (x³ − 1)(x³ − 4)
(iv) (y³ − 2)(y³ + 11)
(v) $\left(a^3-\frac{3}{8}\right)\left(a^3+\frac{16}{17}\right)$
(vi) $\left(a+\frac{3}{4}\right)\left(a+\frac{4}{3}\right)$

Answer 3:

$$\begin{gathered} \left(\mathrm{i}\right) \mathrm{Standard} \mathrm{form} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{polynomial} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}: \\ (5{\mathrm{x}}^4+{\mathrm{x}}^2+6\mathrm{x}+3) \mathrm{or} (3+6\mathrm{x}+{\mathrm{x}}^2+5{\mathrm{x}}^4) \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polynomial} \mathrm{is} 4. \\ \left(\mathrm{ii}\right) \mathrm{Standard} \mathrm{form} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{polynomial} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}: \\ (5{\mathrm{a}}^6+{\mathrm{a}}^2+4) \mathrm{or} (4+{\mathrm{a}}^2+5{\mathrm{a}}^6) \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polynomial} \mathrm{is} 6. \\ \left(\mathrm{iii}\right) ({\mathrm{x}}^3-1)({\mathrm{x}}^3-4)={\mathrm{x}}^6-5{\mathrm{x}}^3+4 \\ \mathrm{Standard} \mathrm{form} \mathrm{of} \mathrm{the} \text{given} \mathrm{polynomial} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}: \\ ({\mathrm{x}}^6-5{\mathrm{x}}^3+4) \mathrm{or} (4-5{\mathrm{x}}^3+{\mathrm{x}}^6) \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polynomial} \mathrm{is} 6. \\ \left(\mathrm{iv}\right) ({\mathrm{y}}^3-2)({\mathrm{y}}^3+11)={\mathrm{y}}^6+9{\mathrm{y}}^3-22 \\ \mathrm{Standard} \mathrm{form} \mathrm{of} \mathrm{the}\text{ given} \mathrm{polynomial} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}: \\ ({\mathrm{y}}^6+9{\mathrm{y}}^3-22) \mathrm{or} (-22+9{\mathrm{y}}^3+{\mathrm{y}}^6) \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polynomial} \mathrm{is} 6. \\ \left(\mathrm{v}\right) ({\mathrm{a}}^3-\frac{3}{8})({\mathrm{a}}^3+\frac{16}{17})={\mathrm{a}}^6+\frac{77}{136}{\mathrm{a}}^3-\frac{6}{17} \\ \mathrm{Standard} \mathrm{form} \mathrm{of} \mathrm{the} \text{given} \mathrm{polynomial} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}: \\ ({\mathrm{a}}^6+\frac{77}{136}{\mathrm{a}}^3-\frac{6}{17}) \mathrm{or} (-\frac{6}{17}+\frac{77}{136}{\mathrm{a}}^3+{\mathrm{a}}^6) \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polynomial} \mathrm{is} 6. \\ \left(\mathrm{vi}\right) (\mathrm{a}+\frac{3}{4})(\mathrm{a}+\frac{4}{3})={\mathrm{a}}^2+\frac{25}{12}\mathrm{a}+1 \\ \mathrm{Standard} \mathrm{form} \mathrm{of} \mathrm{the} \text{given} \mathrm{polynomial} \mathrm{can} \mathrm{be} \mathrm{expressed} \mathrm{as}: \\ ({\mathrm{a}}^2+\frac{25}{12}\mathrm{a}+1) \mathrm{or} (1+\frac{25}{12}\mathrm{a}+{\mathrm{a}}^2) \\ \mathrm{The} \mathrm{degree} \mathrm{of} \mathrm{the} \mathrm{polynomial} \mathrm{is} 2. \end{gathered}$$