Simplifying algebraic fractions

Statement

Simplify the following algebraic fractions: (a) $\dfrac{4x^2 - 16}{12}$; (b) $\dfrac{20 - 30x}{4 - 6x}$; (c) $\dfrac{b^4 - b^2}{b^5 - b^3}$; (d) $\dfrac{12z^5 + 12z^4 + 12z^3}{9z^3 + 9z^2 + 9z}$.

Solution

(a) Factor and use the difference of squares: $\frac{4x^2 - 16}{12} = \frac{4(x^2 - 4)}{12} = \frac{x^2 - 4}{3} = \frac{(x-2)(x+2)}{3}.$

(b) Factor numerator and denominator (with $x \neq \tfrac23$): $\frac{20 - 30x}{4 - 6x} = \frac{10(2 - 3x)}{2(2 - 3x)} = 5.$

(c) Pull out $b^2$ on top and $b^3$ below (with $b \neq 0,\pm1$): $\frac{b^4 - b^2}{b^5 - b^3} = \frac{b^2(b^2 - 1)}{b^3(b^2 - 1)} = \frac{1}{b}.$

(d) Pull out $12z^3$ on top and $9z$ below: $\frac{12z^3(z^2 + z + 1)}{9z(z^2 + z + 1)} = \frac{12z^2}{9} = \frac{4z^2}{3}.$

$\boxed{\dfrac{(x-2)(x+2)}{3}\ ;\ 5\ ;\ \dfrac1b\ ;\ \dfrac{4z^2}{3}}$