# Dividing Polynomials By Monomials

To divide polynomials by monomials, we divide each term of the polynomial in the numerator (dividend) by the monomials in the denominator (divisor) by applying the quotient rule of exponents (am Ã· an = am-n), when needed.

## Dividing a Univariate Polynomial by a Univariate Monomial

Let us divide the polynomial 5p2 – 14p by the monomial p.

Here, we will obtain the quotient by dividing a univariate polynomial by a univariate monomial.

On dividing the terms 5p2 and 14p by p, we get

${\dfrac{5p^{2}}{p}=5p^{2-1}=5p}$ and ${\dfrac{-14p}{p}=-14}$

Thus, the quotient of ${\dfrac{5p^{2}-14p}{p}}$ is ${5p-14}$.

## Dividing a Multivariate Polynomial by a Univariate Monomial

Now, let us divide a multivariate polynomial by a univariate monomial.

On dividing the polynomial 10p2q3 + 12pq4 by the monomial 2q2, we get

${\dfrac{10p^{2}q^{3}+12pq^{4}}{2q^{2}}}$

Dividing the terms 10p2q3 and 12pq4 by 2q2, yields

${\dfrac{10p^{2}q^{3}}{2q^{2}}=\dfrac{10}{2}p^{2}q^{3-2}=5p^{2}q}$

${\dfrac{12pq^{4}}{2q^{2}}=\dfrac{12}{2}pq^{4-2}=6pq^{2}}$

Thus, the quotient is 5p2q + 6pq2.

## Dividing a Multivariate Polynomial by a Multivariate Monomial

Similarly, we can divide a multivariate polynomial 11xyz – 7xy2z3 by the multivariate monomial xyz.

The quotient is:

${\dfrac{11xyz-7xy^{2}z^{3}}{xyz}}$

= ${\dfrac{11xyz}{xyz}-\dfrac{7xy^{2}z^{3}}{xyz}}$

= ${11-7y^{2-1}z^{3-1}}$

= ${11-7yz^{2}}$

Simplify: (18m7 + 12m6 – 24m5) Ã· 6m4

Solution:

${\dfrac{18m^{7}+12m^{6}-24m^{5}}{6m^{4}}}$
= ${\dfrac{18m^{7}+}{6m^{4}}\dfrac{12m^{6}}{6m^{4}}-\dfrac{24m^{5}}{6m^{4}}}$
= ${\dfrac{18}{6}m^{7-4}+\dfrac{12}{6}m^{6-4}-\dfrac{24}{6}m^{5-4}}$
= ${3m^{3}+2m^{2}-4m}$
Thus, (18m7 + 12m6 – 24m5) Ã· 6m4 is simplified to 3m3 + 2m2 – 4m.

## By Factoring

Another method for dividing the polynomials by monomials involves finding their common factors and canceling them.

Let us divide 4u7v3 – 6v7u5 by 2u5v3 using this method.

By finding their common factors, we get

${\dfrac{4u^{7}v^{3}-6v^{7}u^{5}}{2u^{5}v^{3}}}$

= ${\dfrac{2u^{5}v^{3}\left( 2u^{2}-6v^{4}\right) }{2u^{5}v^{3}}}$

Now, by canceling out the common factors from the numerator and the denominator, we get

${2u^{2}-6v^{4}}$

Thus, the quotient is 2u2 – 6v4.

Divide: (-40a8 – 32a7 + 88a11 + 16a2) Ã· 8a2

Solution:

${\dfrac{-40a^{8}-32a^{7}+88a^{11}+16a^{2}}{8a^{2}}}$
= ${\dfrac{8a^{2}\left( -5a^{6}-4a^{5}+11a^{9}+2\right) }{8a^{2}}}$
= ${-5a^{6}-4a^{5}+11a^{9}+2}$
Thus, the quotient is 11a9 – 5a6 – 4a5 + 2.