# Divide and check your answer: 12x^3y^2/2x^5y^2

## Question:

{eq}\displaystyle \frac{12x^3y^2}{2x^5y^2} {/eq}

## Monomials:

The problem we are dealing with comes under polynomial division. Monomials are one type of polynomial. Let's assume two polynomials, A and B. Let A be {eq}\displaystyle x^ay^bz^c {/eq} and polynomial B be {eq}\displaystyle x^my^nz^o {/eq}.

On dividing these polynomials, we get \displaystyle \begin{align} \frac{A}{B} &= \frac{x^ay^bz^c}{x^my^nz^o} \\ &= x^{a-m}y^{b-n}z^{c-o} \end{align}

## Answer and Explanation:

Given:

{eq}\displaystyle \frac{12x^3y^2}{2x^5y^2} {/eq}

On division, we get

{eq}\displaystyle \begin{align} \frac{12x^3y^2}{2x^5y^2} &= \frac{12}{2} \cdot \frac{x^3}{x^5} \cdot \frac{y^2}{y^2} &&\text{[Seperating the denominator and numerator into like terms]} \\ &= 6 x^{3-5} y^{2-2} &&\left[ \frac{a^b}{a^c} = a^{b-c} \right] \\ &= 6x^{-2}y^{0} \\ &= \frac{6}{x^2} &&\left[ x^{-a} = \frac{1}{x^a} \right] \left[ a^{0} = 1 \right] \\ \end{align} {/eq}

Therefore, on dividing the polynomials, we get {eq}\displaystyle \boxed{\mathbf{\frac{6}{x^2}}} {/eq}.