Divide:
(i) $5x^3-15x^2+25x$ by $5x$
(ii) $4z^3+6z^2-z$ by $\frac{-1}{2}z$
(iii) $9x^2y-6xy+12xy^2$ by $\frac{-3}{2}xy$
Given:
The given expressions are:
(i) $5x^3-15x^2+25x$ by $5x$
(ii) $4z^3+6z^2-z$ by $\frac{-1}{2}z$
(iii) $9x^2y-6xy+12xy^2$ by $\frac{-3}{2}xy$
To do:
We have to divide the given expressions.
Solution:
We have to divide the given polynomials by monomials using the formula $x^a \div x^b=a^{a-b}$
Polynomials:
Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.
Monomial:
A monomial is an expression that contains a single term composed of a product of constants and variables with non-negative integer exponents.
Therefore,
(i) The given expression is $5x^3-15x^2+25x$ by $5x$.
$5x^3-15x^2+25x \div 5x=\frac{5x^3}{5x}-\frac{15x^2}{5x}+\frac{25x}{5x}$
$5x^3-15x^2+25x \div 5x=\frac{5}{5}x^{3-1}-\frac{15}{5}x^{2-1}+\frac{25}{5}x^{1-1}$
$5x^3-15x^2+25x \div 5x=x^{2}-3x^{1}+5x^{0}$
$5x^3-15x^2+25x \div 5x=x^{2}-3x+5$ [Since $x^0=1$]
Hence, $5x^3-15x^2+25x$ divided by $5x$ is $x^{2}-3x+5$.
(ii) The given expression is $4z^3+6z^2-z$ by $\frac{-1}{2}z$.
$4z^3+6z^2-z \div \frac{-1}{2}z=\frac{4z^3}{\frac{-1}{2}z}+\frac{6z^2}{\frac{-1}{2}z}-\frac{z}{\frac{-1}{2}z}$
$4z^3+6z^2-z \div \frac{-1}{2}z=(-4\times2)z^{3-1}+(-6\times2)z^{2-1}-(-1\times2)z^{1-1}$
$4z^3+6z^2-z \div \frac{-1}{2}z=-8z^{2}-12z^{1}+2z^{0}$
$4z^3+6z^2-z \div \frac{-1}{2}z=-8z^{2}-12z+2$ [Since $x^0=1$]
Hence, $4z^3+6z^2-z$ divided by $\frac{-1}{2}z$ is $-8z^{2}-12z+2$.
(iii) The given expression is $9x^2y-6xy+12xy^2$ by $\frac{-3}{2}xy$.
$9x^2y-6xy+12xy^2 \div \frac{-3}{2}xy=\frac{9x^2y}{\frac{-3}{2}xy}-\frac{6xy}{\frac{-3}{2}xy}+\frac{12xy^2}{\frac{-3}{2}xy}$
$9x^2y-6xy+12xy^2 \div \frac{-3}{2}xy=(-3\times2)x^{2-1}y^{1-1}-(-2\times2)x^{1-1}y^{1-1}+(-4\times2)x^{1-1}y^{2-1}$
$9x^2y-6xy+12xy^2 \div \frac{-3}{2}xy=-6x^{1}y^{0}-(-4)x^{0}y^{0}+(-8)x^{0}y^{1}$
$9x^2y-6xy+12xy^2 \div \frac{-3}{2}xy=-6x+4-8y$ [Since $x^0=1$]
Hence, $9x^2y-6xy+12xy^2$ divided by $\frac{-3}{2}xy$ is $-6x+4-8y$.
Related Articles
- Multiply:$(2x^2-1)$ by $(4x^3 + 5x^2)$
- 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 $2a$
- Multiply:$(5x + 3)$ by $(7x + 2)$
- Solve the following quadratic equation by factorization: $\frac{2}{x+1}+\frac{3}{2(x-2)}=\frac{23}{5x}, x ≠0, -1, 2$
- Evalute:(i) $(5)^{-2}$(ii) $(-3)^{-2}$(iii) $(\frac{1}{3})^{-4}$(iv) $(\frac{-1}{2})^{-1}$
- Divide $2x^3-5x^2+8x-5 \ by \ 2x^2-3x+5$
- Verify associativity of addition of rational numbers i.e., $(x + y) + z = x + (y + z)$, when:(i) \( x=\frac{1}{2}, y=\frac{2}{3}, z=-\frac{1}{5} \)(ii) \( x=\frac{-2}{5}, y=\frac{4}{3}, z=\frac{-7}{10} \)(iii) \( x=\frac{-7}{11}, y=\frac{2}{-5}, z=\frac{-3}{22} \)(iv) \( x=-2, y=\frac{3}{5}, z=\frac{-4}{3} \)
- On dividing $^{3}+6 z+4 z^{5} $by $ 2 z-1 $, the quotient obtained is a and the remainder is b.What are the values of a and b?A) $ a=2 z^{4}-z^{3}+z^{2}-\frac{z}{2}+\frac{13}{4} \ and \ b=-\frac{13}{4} $B) $ a=2 z^{4}+z^{3}+z^{2}+\frac{z}{2}+\frac{13}{4} \ and \ b=\frac{13}{4} $C) $ a=2 z^{4}-z^{3}+\frac{z^{2}}{2}+\frac{z}{4}+\frac{23}{8} \ and \ b=-\frac{23}{8} $D) $ a=2 z^{4}+z^{3}+\frac{z^{2}}{2}+\frac{z}{4}+\frac{23}{8} \ and \ b=+\frac{23}{8} $
- Expand each of the following, using suitable identities:(i) \( (x+2 y+4 z)^{2} \)(ii) \( (2 x-y+z)^{2} \)(iii) \( (-2 x+3 y+2 z)^{2} \)(iv) \( (3 a-7 b-c)^{2} \)(v) \( (-2 x+5 y-3 z)^{2} \)(vi) \( \left[\frac{1}{4} a-\frac{1}{2} b+1\right]^{2} \)
- Verify that \( x^{3}+y^{3}+z^{3}-3 x y z=\frac{1}{2}(x+y+z)\left[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right] \)
- Solve the following equations and verify your answer:(i) $\frac{7x-2}{5x-1}=\frac{7x+3}{5x+4}$(ii) $(\frac{x+1}{x+2})^2=\frac{x+2}{x+4}$
- Simplify:(i) \( \left(3^{2}+2^{2}\right) \times\left(\frac{1}{2}\right)^{3} \)(ii) \( \left(3^{2}-2^{2}\right) \times\left(\frac{2}{3}\right)^{-3} \)(iii) \( \left[\left(\frac{1}{3}\right)^{-3}-\left(\frac{1}{2}\right)^{-3}\right] \div\left(\frac{1}{4}\right)^{-3} \)(iv) \( \left(2^{2}+3^{2}-4^{2}\right) \div\left(\frac{3}{2}\right)^{2} \)
- Simplify:(i) \( 2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}} \)(ii) \( \left(\frac{1}{3^{3}}\right)^{7} \)(iii) \( \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}} \)(iv) \( 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} \)
- Solve:(i) $3-\frac{2}{5}$(ii) $4+\frac{7}{8}$(iii) $\frac{3}{5}+\frac{2}{7}$(iv) $\frac{9}{11}-\frac{4}{15}$(v) $\frac{7}{10}+\frac{2}{5}+\frac{3}{2}$(vi) $2\frac{2}{3}+3\frac{1}{2}$(vii) $8\frac{1}{2}-3\frac{5}{8}$
- 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 \( \frac{3}{5} b c-\frac{4}{5} a c \)
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