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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 $-6x+4-8y$.
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