Dividing Algebraic Expressions

Division

Similar to multiplication, we handle respective components, divide coefficients and subtract exponents where applicable. We refer to the indices rules again :

$$\frac{x^p}{x^q}=x^{p-q}$$

A key property of fractions is that when the numerator and denominator share a common factor, it cancels out :

$$\frac{ab}{ac}=\frac{\cancel{a}b}{\cancel{a}c}=\frac{b}{c}$$

Let's explore some examples :

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$$\frac{4x^2}{2x}=\frac{2 \cdot2 \cdot x \cdot x}{2 \cdot x}=\frac{\cancel{2} \cdot2 \cdot \cancel{x} \cdot x}{\cancel{2} \cdot \cancel{x}}=\frac{2x}{1}=2x$$

Recall we handle components separately, so the constants divide ($4÷2=2$) and the $x$-terms divide, using indices rules. Here's another way of looking at it :

$$\frac{4x^2}{2x}=\frac{4}{2} \cdot \frac{x^2}{x}=2\cdot x^{2-1}=2x$$

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

$8$ and $3$ don't divide evenly so we can leave those untouched if we want to avoid decimal notation. The $b$-terms are handled using indices rules and the $a$ term doesn't have anything to cancel out with on the bottom of the fraction, so this stays unaffected. Here's a more exhaustive solution :

$$\frac{8b^5a^2}{3b^2}=\frac{8}{3} \cdot \frac{b^5}{b^2} \cdot \frac{a^2}{1}=\frac{8}{3} \cdot b^{5-2} \cdot a^2=\frac{8b^3a^2}{3}$$

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In many cases, expressions may need to be factored so we can identify common factors on the denominator and numerator.

Example