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 (a^m ÷ aⁿ = a^m-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:

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

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

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

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

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.