Division of Algebraic Terms

This lesson covers how to divide algebraic terms. The method mirrors multiplication: fix the sign, divide the number coefficients, then subtract the exponents of each repeated variable.

Sign rules for division

When dividing signed numbers, the sign of the quotient follows four rules:

$\frac{(+)}{(+)} = +,\quad \frac{(-)}{(-)} = +,\quad \frac{(+)}{(-)} = -,\quad \frac{(-)}{(+)} = -$

Law of exponents

Dividing powers of the same base subtracts the exponents:

$\frac{a^{m}}{a^{n}} = a^{m-n}$

This is the rule used throughout the examples.

Example 1: $\dfrac{30x^{2}y^{2}}{15xy}$

Divide the numbers, then subtract the powers of each variable.

$\frac{30x^{2}y^{2}}{15xy} = \frac{30}{15}\cdot\frac{x^{2}}{x}\cdot\frac{y^{2}}{y}$

Numbers: $30\div 15 = 2$. Variable $x$: $x^{2-1} = x$. Variable $y$: $y^{2-1} = y$.

$= 2xy$

Example 2: $\dfrac{-80p^{3}q}{-2p^{2}q}$

First the sign: negative divided by negative is positive.

$\frac{-80p^{3}q}{-2p^{2}q} = \frac{-80}{-2}\cdot\frac{p^{3}}{p^{2}}\cdot\frac{q}{q}$

Numbers: $\tfrac{-80}{-2} = 40$. Variable $p$: $p^{3-2} = p$. Variable $q$: $q^{1-1} = q^{0} = 1$.

$= 40p$

Example 3: $\dfrac{3xy^{3}z}{2xyz}$

$\frac{3xy^{3}z}{2xyz} = \frac{3}{2}\cdot\frac{x}{x}\cdot\frac{y^{3}}{y}\cdot\frac{z}{z}$

Numbers: $\tfrac{3}{2}$ does not reduce. Variable $x$: $x^{1-1} = 1$. Variable $y$: $y^{3-1} = y^{2}$. Variable $z$: $z^{1-1} = 1$.

$= \tfrac{3}{2}y^{2}$

Example 4: $\dfrac{14a^{2}b^{2}}{2ab}\times\dfrac{3a^{2}b}{3ab}$

Simplify each quotient first, then multiply the results.

Left quotient: $\dfrac{14a^{2}b^{2}}{2ab} = 7a^{2-1}b^{2-1} = 7ab$.

Right quotient: $\dfrac{3a^{2}b}{3ab} = a^{2-1}b^{1-1} = a$.

Now multiply, adding the powers of $a$:

$7ab \times a = 7a^{1+1}b = 7a^{2}b$

Example 5: $\dfrac{-70x^{3}y^{2}}{10x^{2}y}$

$\frac{-70x^{3}y^{2}}{10x^{2}y} = \frac{-70}{10}\cdot\frac{x^{3}}{x^{2}}\cdot\frac{y^{2}}{y}$

Numbers: $\tfrac{-70}{10} = -7$. Variable $x$: $x^{3-2} = x$. Variable $y$: $y^{2-1} = y$.

$= -7xy$

Example 6: $\dfrac{40x^{2}y^{2} + 10x^{2}y^{2}}{8x^{2}y^{2} - 5x^{2}y^{2}}$

Here there is an addition in the numerator and a subtraction in the denominator, then a division. Combine the like terms first.

Numerator: $40x^{2}y^{2} + 10x^{2}y^{2} = 50x^{2}y^{2}$. Denominator: $8x^{2}y^{2} - 5x^{2}y^{2} = 3x^{2}y^{2}$.

$\frac{50x^{2}y^{2}}{3x^{2}y^{2}} = \frac{50}{3}\cdot\frac{x^{2}}{x^{2}}\cdot\frac{y^{2}}{y^{2}} = \tfrac{50}{3}$

The fraction $\tfrac{50}{3}$ does not divide exactly, so it is left as it is.

Example 7: Area of a square with side $4pq$

The area of a square is side times side.

$\text{Area} = 4pq \times 4pq = (4\times 4)\,p^{2}\,q^{2} = 16p^{2}q^{2}\ \text{square units}$

Example 8: $2pq\times 5qr + (-3pq^{2})\times 2r$

Multiply each product, then combine the like terms.

First product: $2pq \times 5qr = 10pq^{2}r$. Second product: $-3pq^{2}\times 2r = -6pq^{2}r$.

Both terms are $pq^{2}r$, so they are like terms:

$10pq^{2}r - 6pq^{2}r = 4pq^{2}r$

Example 9: $(2abc + 4abc)\times\big(12abc - (-3abc)\big)$

Simplify each bracket first.

First bracket: $2abc + 4abc = 6abc$. Second bracket: $12abc - (-3abc) = 12abc + 3abc = 15abc$.

Now multiply:

$6abc \times 15abc = (6\times 15)\,a^{2}\,b^{2}\,c^{2} = 90a^{2}b^{2}c^{2}$

Key takeaways

• The sign of a quotient follows the same rules as multiplication: like signs give a positive result, unlike signs give a negative result.
• Divide the numerical coefficients, and leave the fraction as it is when it does not reduce to a whole number.
• For each variable, subtract the exponent in the denominator from the exponent in the numerator.