Which of the following expressions are not polynomials?:
(i) $x^2+2x^{-2}$
(ii) $\sqrt{ax}+x^2-x^3$
(iii) $3y^3-\sqrt{5}y+9$
(iv) $ax^{\frac{1}{2}}y^7+ax+9x^2+4$
(v) $3x^{-3}+2x^{-1}+4x+5$
Given:
The given expressions are:
(i) $x^2+2x^{-2}$
(ii) $\sqrt{ax}+x^2-x^3$
(iii) $3y^3-\sqrt{5}y+9$
(iv) $ax^{\frac{1}{2}}y^7+ax+9x^2+4$
(v) $3x^{-3}+2x^{-1}+4x+5$
To do:
We have to find which of the given expressions are polynomials.
Solution:
Polynomials:
Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.
To identify whether the given expression is polynomial, check if all the powers of the variables are whole numbers after simplification. If any of the powers is a fraction or negative integer then it is not a polynomial.
(i) The given expression is $x^2+2x^{-2}$.
The term $2x^{-2}$ has a negative power of $-2$.
Therefore,
The given expression is not a polynomial.
(ii) The given expression is $\sqrt{ax}+x^2-x^3$.
The term $\sqrt{ax}=\sqrt{a}x^{\frac{1}{2}}$ has a fraction power of $\frac{1}{2}$.
Therefore,
The given expression is not a polynomial.
(iii) The given expression is $3y^3-\sqrt{5}y+9$.
The given expression does not have any negative or fractional powers.
Therefore,
The given expression is a polynomial.
(iv) The given expression is $ax^{\frac{1}{2}}y^7+ax+9x^2+4$
The term $ax^{\frac{1}{2}}y^7$ has a fraction power of $\frac{1}{2}$.
Therefore,
The given expression is not a polynomial.
(v) The given expression is $3x^{-3}+2x^{-1}+4x+5$.
The terms $3x^{-3}$ and $2x^{-1}$ have negative powers of $-2$ and $-1$.
Therefore,
The given expression is not a polynomial.
Related Articles
- Find the following products:(i) $(x + 4) (x + 7)$(ii) $(x - 11) (x + 4)$(iii) $(x + 7) (x - 5)$(iv) $(x - 3) (x - 2)$(v) $(y^2 - 4) (y^2 - 3)$(vi) $(x + \frac{4}{3}) (x + \frac{3}{4})$(vii) $(3x + 5) (3x + 11)$(viii) $(2x^2 - 3) (2x^2 + 5)$(ix) $(z^2 + 2) (z^2 - 3)$(x) $(3x - 4y) (2x - 4y)$(xi) $(3x^2 - 4xy) (3x^2 - 3xy)$(xii) $(x + \frac{1}{5}) (x + 5)$(xiii) $(z + \frac{3}{4}) (z + \frac{4}{3})$(xiv) $(x^2 + 4) (x^2 + 9)$(xv) $(y^2 + 12) (y^2 + 6)$(xvi) $(y^2 + \frac{5}{7}) (y^2 - \frac{14}{5})$(xvii) $(p^2 + 16) (p^2 - \frac{1}{4})$
- \Find $(x +y) \div (x - y)$. if,(i) \( x=\frac{2}{3}, y=\frac{3}{2} \)(ii) \( x=\frac{2}{5}, y=\frac{1}{2} \)(iii) \( x=\frac{5}{4}, y=\frac{-1}{3} \)(iv) \( x=\frac{2}{7}, y=\frac{4}{3} \)(v) \( x=\frac{1}{4}, y=\frac{3}{2} \)
- Solve the following pairs of equations by reducing them to a pair of linear equations:(i) \( \frac{1}{2 x}+\frac{1}{3 y}=2 \)\( \frac{1}{3 x}+\frac{1}{2 y}=\frac{13}{6} \)(ii) \( \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2 \)\( \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1 \)(iii) \( \frac{4}{x}+3 y=14 \)\( \frac{3}{x}-4 y=23 \)(iv) \( \frac{5}{x-1}+\frac{1}{y-2}=2 \)\( \frac{6}{x-1}-\frac{3}{y-2}=1 \)(v) \( \frac{7 x-2 y}{x y}=5 \)\( \frac{8 x+7 y}{x y}=15 \),b>(vi) \( 6 x+3 y=6 x y \)\( 2 x+4 y=5 x y \)4(vii) \( \frac{10}{x+y}+\frac{2}{x-y}=4 \)\( \frac{15}{x+y}-\frac{5}{x-y}=-2 \)(viii) \( \frac{1}{3 x+y}+\frac{1}{3 x-y}=\frac{3}{4} \)\( \frac{1}{2(3 x+y)}-\frac{1}{2(3 x-y)}=\frac{-1}{8} \).
- Which of the following expressions are polynomials? Which are not? Give reasons.(i) \( 4 x^{2}+5 x-2 \)(ii) \( y^{2}-8 \)(iii) 5(iv) \( 2 x^{2}+\frac{3}{x}-5 \)
- Determine which of the following polynomials has \( (x+1) \) a factor:(i) \( x^{3}+x^{2}+x+1 \)(ii) \( x^{4}+x^{3}+x^{2}+x+1 \)(iii) \( x^{4}+3 x^{3}+3 x^{2}+x+1 \)(iv) \( x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2} \)
- Add the following algebraic expressions(i) \( 3 a^{2} b,-4 a^{2} b, 9 a^{2} b \)(ii) \( \frac{2}{3} a, \frac{3}{5} a,-\frac{6}{5} a \)(iii) \( 4 x y^{2}-7 x^{2} y, 12 x^{2} y-6 x y^{2},-3 x^{2} y+5 x y^{2} \)(iv) \( \frac{3}{2} a-\frac{5}{4} b+\frac{2}{5} c, \frac{2}{3} a-\frac{7}{2} b+\frac{7}{2} c, \frac{5}{3} a+ \) \( \frac{5}{2} b-\frac{5}{4} c \)(v) \( \frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x,-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y \)(vi) \( \frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}, \frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3} \) \( \frac{3}{2} x^{2}-\frac{5}{2} x-2 \)
- Divide:(i) $x+2x^2+3x^4-x^5$ by $2x$(ii) $y^4-3y^3+\frac{1}{2y^2}$ by $3y$(iii) $-4a^3+4a^2+a$ by $2a$
- Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.(i) \( 4 x^{2}-3 x+7 \)(ii) \( y^{2}+\sqrt{2} \)(iii) \( 3 \sqrt{t}+t \sqrt{2} \)(iv) \( y+\frac{2}{y} \)(v) \( x^{10}+y^{3}+t^{50} \)
- Factorise:(i) \( x^{3}-2 x^{2}-x+2 \)(ii) \( x^{3}-3 x^{2}-9 x-5 \)(iii) \( x^{3}+13 x^{2}+32 x+20 \)(iv) \( 2 y^{3}+y^{2}-2 y-1 \)
- Write the degree of each of the following polynomials:(i) \( 5 x^{3}+4 x^{2}+7 x \)(ii) \( 4-y^{2} \)(iii) \( 5 t-\sqrt{7} \)(iv) 3
- Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.$( i).\ 4x^{2}-3x+7$$( ii).\ y^{2}+\sqrt{2}$$( iii).\ 3\sqrt{t}+t\sqrt{2}$$( iv).\ x^{10}+y^{3}+t^{10}$$( v).\ y+\frac{2}{y}$
- Solve the following pair of linear equations by the substitution method.(i) $x + y = 14, x – y = 4$(ii) $s – t = 3, \frac{s}{3} + \frac{t}{2} = 6$(iii) $3x – y = 3, 9x – 3y = 9$(iv) $0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3$(v) \( \sqrt{2} x+\sqrt{3} y=0, \sqrt{3} x-\sqrt{8} y=0 \)(vi) \( \frac{3 x}{2}-\frac{5 y}{3}=-2, \frac{x}{3}+\frac{y}{2}=\frac{13}{6} \).
- (i) \( x^{2}-3 x+5-\frac{1}{2}\left(3 x^{2}-5 x+7\right) \)(ii) \( [5-3 x+2 y-(2 x-y)]-(3 x-7 y+9) \)(iii) \( \frac{11}{2} x^{2} y-\frac{9}{4} x y^{2}+\frac{1}{4} x y-\frac{1}{14} y^{2} x+\frac{1}{15} y x^{2}+ \) \( \frac{1}{2} x y \)(iv) \( \left(\frac{1}{3} y^{2}-\frac{4}{7} y+11\right)-\left(\frac{1}{7} y-3+2 y^{2}\right)- \) \( \left(\frac{2}{7} y-\frac{2}{3} y^{2}+2\right) \)(v) \( -\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+ \) \( \frac{1}{6} c b^{2} a+\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b \).
- Subtract:(i) $-5xy$ from $12xy$(ii) $2a^2$ from $-7a^2$(iii) \( 2 a-b \) from \( 3 a-5 b \)(iv) \( 2 x^{3}-4 x^{2}+3 x+5 \) from \( 4 x^{3}+x^{2}+x+6 \)(v) \( \frac{2}{3} y^{3}-\frac{2}{7} y^{2}-5 \) from \( \frac{1}{3} y^{3}+\frac{5}{7} y^{2}+y-2 \)(vi) \( \frac{3}{2} x-\frac{5}{4} y-\frac{7}{2} z \) from \( \frac{2}{3} x+\frac{3}{2} y-\frac{4}{3} z \)(vii) \( x^{2} y-\frac{4}{5} x y^{2}+\frac{4}{3} x y \) from \( \frac{2}{3} x^{2} y+\frac{3}{2} x y^{2}- \) \( \frac{1}{3} x y \)(viii) \( \frac{a b}{7}-\frac{35}{3} b c+\frac{6}{5} a c \) from \( \frac{3}{5} b c-\frac{4}{5} a c \)
- If $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}=x,\ \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}=y$, find the value $x^{2}+y^{2}+x y$.
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies