\(\displaystyle \frac{(12w^{2}-15w+21w^{3})}{3w}\)

When dividing a polynomial by a monomial, we can use a divison called, term-by-term, dividing each of the top terms by the monomial.

\(\displaystyle \frac{12w^{2}}{3w}-\frac{15w}{3w}+\frac{21w^{3}}{3w}\)

Simplify.

\(\displaystyle 4w-5+7w^{2}\)

Rewrite it with the leading coefficent first, 

Final Answer: \(\displaystyle 7w^{2}+4w-5\)

The first step is to divide the constants, 18 and 6, by the LCM, 6, to get 3. When dividing variables, if the variable is present in the numerator and denominator, subtract the exponent found in the numerator by the exponent in the denominator.

For \(\displaystyle x\), you have \(\displaystyle 4-1=3\).

For \(\displaystyle y\), you have \(\displaystyle 3-2=1\).

For \(\displaystyle z\), you have \(\displaystyle 1-3=-2\).

Then write the simplified answer as one term:

\(\displaystyle 3x^3yz^{-2}\)

\(\displaystyle \frac{9x^4+6x^3-12x}{3x}=\frac{9x^4}{3x}+\frac{6x^3}{3x}-\frac{12x}{3x}=3x^3+2x^2-4\)

For any polynomial division, divide each term in the numerator individually by the denominator:

\(\displaystyle \frac{2x^{2}y^{3}z - 6xy^{2}z^{2}+10x^{3}yz^{3}}{2xyz} =\) \(\displaystyle \frac{2x^{2}y^{3}z}{2xyz} - \frac{6xy^{2}z^{2}}{2xyz} + \frac{10x^{3}yz^{3}}{2xyz}=\)

\(\displaystyle xy^{2}-3yz+5x^{2}z^{2}\)

When dividing monomials, consider the coefficients and variables separately. Rewrite the expression as  \(\displaystyle \left(\frac{14}{7}\right)\left(\frac{x^2}{x}\right)\left(y\right)\), grouping common bases. For the coeffiecients, we can divide normally: \(\displaystyle \left(\frac{14}{7} \right )=2\). For the variables, we can keep the common base and subtract the exponents: \(\displaystyle \left(\frac{x^2}{x} \right )=x^{2-1}=x\). Then, multiply each portion all back together to obtain \(\displaystyle 2xy\).

\(\displaystyle \small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}\)

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

\(\displaystyle \small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}=\frac{x^2y^2x^3y^4z}{x^4y^3z^4}\)

To combine like terms in the numerator, we add their exponents.

\(\displaystyle \small \frac{x^5y^6z}{x^4y^3z^4}\)

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

\(\displaystyle x^{5-4}y^{6-3}z^{1-4}=xy^3z^{-3}\)

Remember that any negative exponents stay in the denominator.

\(\displaystyle \frac{xy^3}{z^3}\)

\(\displaystyle p^{7}\) and \(\displaystyle p^{4}\) cancel out, leaving \(\displaystyle p^{3}\) in the numerator. 5 and 25 cancel out, leaving 5 in the denominator

\(\displaystyle \frac{4x+6x^2}{2x}\)

First, let us factor the numerator:

\(\displaystyle 4x+6x^2=x(4+6x)=2x(2+3x)\)

\(\displaystyle \frac{2x(2+3x)}{2x}=2+3x\)

First, flip the numerator and the denominator of the second fraction to turn the division into multiplication.

\(\displaystyle \frac{-2pl^2}{3yz}\div \frac{9lp^2}{5xy}=\frac{-2pl^2}{3yz} * \frac{5xy}{9lp^2}=\frac{-2pl^25xy}{3yz9lp^2}\)

We can then cancel like terms.

From both the numerator and denominator, remove one \(\displaystyle p\), remove one \(\displaystyle l\), and remove one \(\displaystyle y\):

\(\displaystyle \frac{-2l5x}{3z9p}\)

Then we finish by multiplying the constants:

\(\displaystyle \frac{-10lx}{27zp}\)

When different powers of the same variable are multiplied, the exponents are added.  When different powers of the same variable are divided, the exponents are subtracted.  So, as an example:

\(\displaystyle \frac{x^{3}}{x} = x^{\left ( 3-1\right )} = x^{2}\)

For the above problem, 

\(\displaystyle \frac{x^{2}}{x^{3}} = \frac{1}{x} ; \frac{y^{-3}}{y^{-2}} = \frac{y^{2}}{y^{3}} = \frac{1}{y} ; \frac{z^{4}}{z^{2}} = z^{2}\)

Therefore, the expression simplifies to:

\(\displaystyle \left ( \frac{1}{x} \right ) \left ( \frac{1}{y} \right ) z^{2} = \frac{z^{2}}{xy}\)