Class 8 Division of Algebraic Expressions Exercise 8.1

\begin{array}{l} \text {Q1. Write the degree of each of the following polynomials: } \\ \text {(i) } \quad 2 x^{3}+5 x^{2}-7 \\ \text {(ii) } \quad 5 x^{2}-3 x+2 \\ \text {(iii) } \quad 2 x+x^{2}-8 \\ \text {(iv) } \quad \frac {1}{2} y^{7}-12 y^{6}+48 y^{5}-10 \\ \text {(v) } \quad 3 x^{3}+1 \\ \text {(vi) } \quad 5 \\ \text {(vii) } \quad 20 x^{3}+12 x^{2} y^{2}-10 y^{2}+20 \\ \\\text {Sol. We know that in a polynomial, degree is the highest power of the variable.} \\ \text {(i) } \quad 2 x^{3}+5 x^{2}-7 \\\text {The degree of the polynomial } 2 x^{3}+5 x^{2}-7 \text { is 3.} \\ \\ \text {(ii) } \quad 5 x^{2}-3 x+2 \\ \text {The degree of the polynomial } 5 x^{2}-3 x+2 is \text { 2.} \\ \\\text {(iii) } \quad 2 x+x^{2}-8 \\ \text {The degree of the polynomial } 2 x+x^{2}-8 \text { is 2.} \\ \\\text {(iv) } \quad \frac {1}{2}y^{7}-12 y^{6}+48 y^{5}-10 \\ \text {The degree of the polynomial } \frac {1}{2} y^{7}-12 y^{6}+48 y^{5}-10 \text { is 7.} \\ \\\text {(v) } \quad 3 x^{3}+1 \\ \text {The degree of the polynomial } 3 x^{3}+1 \text { is 3.} \\ \\\text {(vi) } \quad 5 \\ \text {The degree of the polynomial 5 is 0.} \quad [\because \text {5 is a constant number}] \\ \\\text {(vii) } \quad 20 x^{3}+12 x^{2} y^{2}-10 y^{2}+20 \\ \text {The degree of the polynomial } 20 x^{3}+12 x^{2} y^{2}-10 y^{2}+20 \text {is 4.} \\ \\\text {Q2. Which of the following expressions are not polynomials?} \\ \text {(i) } \quad x^{2}+2 x^{-2} \\ \text {(ii) } \quad \sqrt{a}x +x^{2}-x^{3} \\ \text {(iii) } \quad 3 y^{3}-\sqrt{5}y+9 \\ \text {(iv) } \quad a x^{\frac {1}{2}}+a x+9 x^{2}+4 \\ \text {(v) } \quad 3 x^{-3}+2 x^{-1}+4 x+5 \\ \\\text {(i) } \quad x^{2}+2 x^{-2} \\ \text {It’s not a polynomial because a polynomial does not contain any negative or fraction power.} \\ \text {And here x is having power of -2 which is a negative integrer.} \\ \\\text {(ii) } \quad \sqrt{a}x+x^{2}-x^{3} \\ \text {It’s a polynomial because powers of the variable x is non-negative integers.} \\ \\\text {(iii) } \quad 3 y^{3}-\sqrt{5}y+9 \\ \text {It’s a polynomial because powers of the variable x is non-negative integers.} \\ \\\text {(iv) } \quad a x^{\frac {1}{2}}+a x+9 x^{2}+4 \\ \text {It’s not a polynomial because power of variable x is in fraction i.e } \frac {1}{2} \\ \\\text {(v) } \quad 3 x^{-3}+2 x^{-1}+4 x+5 \\ \text {It’s not a polynomial because power of variable x is in negative.} \\ \\\text {Q3. Write each of the following polynomials in the standard from. Also, write their degree: } \\ \text {(i) } \quad x^{2}+3+6 x+5 x^{4} \\ \text {(ii) } \quad a^{2}+4+5 a^{6} \\ \text {(iii) } \quad (x^{3}-1)(x^{3}-4) \\ \text {(iv) } \quad (y^{3}-2)(y^{3}+11) \\ \text {(v) } \quad (a^{3}-\frac {3}{8})(a^{3}+ \frac {16}{17}) \\ \text {(vi) } \quad (a + \frac {3}{4})(a + \frac {4}{3}) \\ \\\text {Sol. The standard form of the polynomial is written in either increasing or } \\ \text {decreasing order of their powers.} \\\text {(i) } \quad x^{2}+3+6x+5 x^{4} \\ \text {The standard form is } \\ (5 x^{4}+x^{2}+6 x+3) \text { or } (3+6 x+x^{2}+5 x^{4}) \\\text {The degree of the polynomial is 4.} \\ \\\text {(ii) } \quad a^{2}+4+5 a^{6} \\ \text {The standard form is } \\ (5 a^{6}+a^{2}+4) \text { or } (4+a^{2}+5 a^{6}) \\ \text {The degree of the polynomial is 6.} \\ \\\text {(iii) } \quad (x^{3}-1)(x^{3}-4)=x^{6}-5 x^{3}+4 \\\text {The standard form is } \\ (x^{6}-5 x^{3}+4) \text { or } (4-5 x^{3}+x^{6}) \\ \text {The degree of the polynomial is 6.} \\ \\\text {(iv) } \quad (y^{3}-2)(y^{3}+11) =y^{6}+9 y^{3}-22 \\ \text {The standard form is } \\ (y^{6}+9 y^{3}-22) \text { or } (-22+9 y^{3}+y^{6}) \\ \text {The degree of the polynomial is 6.} \\ \\\text {(v) } \quad (a^{3}-\frac {3}{8})(a^{3}+ \frac {16}{17}) \\ =a^{6}+\frac{27}{136} a^{3}-\frac{48}{136} \\ \text {The standard form is } \\ (a^{6}+\frac{27}{136} a^{3}-\frac{48}{136}) \text { or } (-\frac{48}{136}+\frac{27}{136} a^{3}+a^{6}) \\ \text {The degree of the polynomial is 6.} \\ \\\text {(vi) } \quad (a + \frac {3}{4})(a + \frac {4}{3}) =a^{2}+\frac{25}{12} a+1\\ \text {The standard form is } \\(a^{2}+\frac{25}{12} a+1) \text { or } (1+\frac{25}{12} a+a^{2}) \\ \text {The degree of the polynomial is 2.} \\ \\\end{array}