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Dividing by a Monomial Solutions

Division can be written in a number of ways.
For example, “Divide $A x^{2} y^{2} + B x y + C y$ by Axy” can be written symbolically as any of these options seen here:

$(A x^{2} y^{2} + B x y + C y) {(A x y)}^{- 1}$

$\frac{A x^{2} y^{2} + B x y + C y}{A x y}$

$(A x^{2} y^{2} + B x y + C y) \div (A x y)$

$A x y A x^{2} y^{2} + B x y + C y$

Note

The $A x y A x^{2} y^{2} + B x y + C y$ notation for dividing monomials is less common, but used when the divisor is a binomial or greater number of terms.

The dividend is the product of the divisor and quotient (plus the remainder, if present).
A remainder is present when the divisor does not go into the dividend evenly.
Use the exponent rules when dividing a polynomial by a monomial.
Divide every term in the polynomial (dividend) by the monomial (divisor).
It may be helpful to think of the monomial as the least common denominator (LCD) of each term.
Simplify the coefficients like numerical fractions.
Simplify the variables using the exponent rules.

Example 1

Simplify.
$(9 a^{4} b^{2} - 15 a^{3} b^{2} + 8 a^{2} b - 6 a^{2}) {(3 a b)}^{- 1}$

Implement

$\frac{9 a^{4} b^{2} - 15 a^{3} b^{2} + 8 a^{2} b - 6 a^{2}}{3 a b}$

$\frac{9 a^{4} b^{2}}{3 a b} - \frac{15 a^{3} b^{2}}{3 a b} + \frac{8 a^{2} b}{3 a b} - \frac{6 a^{2}}{3 a b}$

$3 a^{3} b - 5 a^{2} b + \frac{8}{3} a - \frac{2 a}{b}$

Explain

Rewrite expression as a fraction
Rewrite the expression using the monomial as the LCD
Simplify each term (fractional coefficients and exponent rules)

Example 2

Simplify.
$\frac{5 x^{3} y^{2} z + 6 x^{2} y z - x y z}{x y z}$

Implement

$\frac{5 x^{3} y^{2} z}{x y z} + \frac{6 x^{2} y z}{x y z} - \frac{x y z}{x y z}$

$5 x^{2} y + 6 x - 1$

Explain

Rewrite the expression using the monomial as the LCD
Simplify each term

Note

It can be helpful to rewrite the problem with the LCD under each term, checking that all terms are simplified correctly.

Alternatively, you could factor out the GCF; however, this method would require that you consider fractional coefficients.