Factor with Fractions

We present examples on how to use the properties of commutativity, associativity and distributivity and the different rules of fractions to factor expressions including fractions.
Questions and their answers are also included.   $\require{cancel}$ $\newcommand\ccancel[2][black]{\color{#1}{\xcancel{\color{black}{#2}}}}$ A review of the properties of commutativity, associativity and distributivity and the different rules of fractions may be needed before you start the examples and questions below.

Application of the Properties to Factor with Fractions

Example 1
Use distributivity to factor completely the following expressions.
a) $\quad \dfrac{x}{4} + \dfrac{1}{2}$

b) $\quad \dfrac{4}{3} \times \dfrac{x}{2} + \dfrac{1}{6}$

c) $\quad \dfrac{3x}{16} + \dfrac{9}{8}$

c) $\quad \dfrac{2 x^2}{7} + \dfrac{4 x }{21}$

Solution to Example 1 To factor fractions, we first look at a greatest common factors (GCF) in the numerators and a GCF in the denominators.
a)
Given: $\dfrac{x}{4} + \dfrac{1}{2}$
The numerators $x$ and $1$ of the two fractions have $1$ as a common factor. The denominators $4$ and $2$ have a greatest common factor equal to $2$. Hence the fraction $\dfrac{1}{2}$ is a common factor to the two terms in the given expression.
Write each of the fractions $\dfrac{x}{4}$ and $\dfrac{1}{2}$ as a product of the common factor $\dfrac{1}{2}$ and another fraction
$\quad\quad \dfrac{x}{4} + \dfrac{1}{2} = \dfrac{1}{2} \times \dfrac{x}{2} + \dfrac{1}{2} \times \dfrac{1}{1}$
Use ditsributivity (from right to left) to factor the fraction $\dfrac{1}{2}$ out
$\quad\quad= \dfrac{1}{2} \left(\dfrac{x}{2} + 1 \right)$

b)
Given: $\dfrac{4}{3} \times \dfrac{x}{2} + \dfrac{1}{6}$
Multiply the fractions on the left
$\quad\quad \dfrac{4}{3} \times \dfrac{x}{2} + \dfrac{1}{6} = \dfrac{4 x}{6} + \dfrac{1}{6}$
The numerators $4x$ and $1$ of the two fractions above have $1$ as common factor. However the denominators are equal to $6$. Therefore the fraction $\dfrac{1}{6}$.
Write each of the fractions $\dfrac{4 x}{6}$ and $\dfrac{1}{6}$ as a product of the common factor $\dfrac{1}{6}$ and another fraction
$\quad\quad = \dfrac{1}{6} \times \dfrac{4 x}{1} + \dfrac{1}{6} \times \dfrac{1}{1}$
Factor the fraction $\dfrac{1}{6}$ out
$\quad\quad = \dfrac{1}{6} \left ( 4 x + 1 \right )$

c)
Given: $\dfrac{3x}{16} + \dfrac{9}{8}$
The numerators $3x$ and $9$ of the two fractions above have a greaest common factor equal to $3$. The denominators have a greatest common factor equal to $8$. Therefore the fraction $\dfrac{3}{8}$ is a common factor.
Write each of the fractions $\dfrac{3x}{16}$ and $\dfrac{9}{8}$ as a product of the common factor $\dfrac{3}{8}$ and another fraction
$\quad\quad = \dfrac{3}{8} \times \dfrac{x}{2} + \dfrac{3}{8} \times \dfrac{3}{1}$
Factor the fraction $\dfrac{3}{8}$ out
$\quad\quad = \dfrac{3}{8} \left ( \dfrac{x}{2} + 3 \right )$

d)
Given: $\dfrac{2 x^2}{7} + \dfrac{4 x }{21}$
The numerators $2 x^2$ and $4 x$ of the two fractions above have a common factor equal to $2 x$. The denominators have a greatest common fcator equal to $7$. Therefore the fraction $\dfrac{2x}{7}$ is a common factor.
Write each of the fractions $\dfrac{2 x^2}{7}$ and $\dfrac{4 x }{21}$ as a product of the common factor $\dfrac{2x}{7}$ and another fraction
$\quad\quad = \dfrac{2x}{7} \times \dfrac{x}{1} + \dfrac{2x}{7} \times \dfrac{2}{3}$
Factor out the fraction $\dfrac{2x}{7}$
$\quad\quad = \dfrac{2x}{7} \left ( x + \dfrac{2}{3} \right)$

Questions

Factor completely the following expressions.

1. ) $\dfrac{2x}{3} + \dfrac{4}{9}$

2. ) $\dfrac{8}{3x} - \dfrac{4}{9}$

3. ) $\dfrac{9x^2}{4} + \dfrac{3x}{8}$

4. ) $\dfrac{ 5 x y}{7} + \dfrac{10 y}{21}$

1. ) $\dfrac{2x}{3} + \dfrac{4}{9} = \dfrac{2}{3} \left( x + \dfrac{2}{3} \right)$

2. ) $\dfrac{8}{3x} - \dfrac{4}{9} = \dfrac{4}{3} \left( \dfrac{2}{x} - \dfrac{1}{3} \right)$

3. ) $\dfrac{9x^2}{4} + \dfrac{3x}{8} = \dfrac{3x}{4} \left( 3 x + \dfrac{1}{2} \right)$

4. ) $\dfrac{ 5 x y}{7} + \dfrac{10 y}{21} = \dfrac{5 y}{7} \left( \dfrac{x}{7} + \dfrac{2}{3} \right)$