Factoring a Monomial

1. Decompose the coefficient into prime factors.

2. Decompose the variables.

3. Write the monomial as a product of prime factors.

Factor the monomial |300x^3yz^2.|

1. Decompose the coefficient into prime factors
   Several techniques can be used to find the prime factorization. The factor tree is one of them.
   ||300=2\times 2\times 3\times 5\times 5||

2. Decompose the variables
   ||\color{#333FB1}{x^3}\color{#EC0000}{y}\color{#3A9A38}{z^2}=\color{#333FB1}{x}\times \color{#333FB1}{x}\times \color{#333FB1}{x}\times \color{#EC0000}{y}\times \color{#3A9A38}{z}\times \color{#3A9A38}{z}||

3. Write the monomial as a product of prime factors
   ||300x^3yz^2=2\times 2\times 3\times 5\times 5 \times x\times x\times x\times y\times z\times z||

Factoring a monomial is useful when simplifying a fraction that contains a monomial in the numerator and the denominator; thus, the same steps are performed twice.

Simplify the fraction |\dfrac{18a^4b^3c}{6a^3bc^2}.|

1. Decompose the coefficients into prime factors
   ||\begin{align}\color{#333FB1}{18}&=\color{#333FB1}{2}\times \color{#333FB1}{3}\times \color{#333FB1}{3}\\ \color{#333FB1}{6}&=\color{#333FB1}{2}\times \color{#333FB1}{3}\end{align}||

2. Decompose the variables
   ||\begin{align}\color{#3A9A38}{a^4}\color{#EC0000}{b^3}\color{#FA7921}{c}&=\color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#EC0000}{b}\times \color{#EC0000}{b}\times \color{#EC0000}{b}\times \color{#FA7921}{c}\\ \color{#3A9A38}{a^3}\color{#EC0000}{b}\color{#FA7921}{c^2}&=\color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#EC0000}{b}\times \color{#FA7921}{c}\times \color{#FA7921}{c}\end{align}||

3. Write each monomial as a product of prime factors
   ||\dfrac{\color{#333FB1}{18}\color{#3A9A38}{a^4}\color{#EC0000}{b^3}\color{#FA7921}{c}}{\color{#333FB1}{6}\color{#3A9A38}{a^3}\color{#EC0000}{b}\color{#FA7921}{c^2}}=\dfrac{\color{#333FB1}{2}\times \color{#333FB1}{3}\times \color{#333FB1}{3}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#EC0000}{b}\times \color{#EC0000}{b}\times \color{#EC0000}{b}\times \color{#FA7921}{c}}{\color{#333FB1}{2}\times \color{#333FB1}{3}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#3A9A38}{a}\times \color{#EC0000}{b}\times \color{#FA7921}{c}\times \color{#FA7921}{c}}||

4. Simplify the fraction by eliminating common factors
   ||\begin{align} &\dfrac{\cancel{\color{#333FB1}{2}}\times \cancel{\color{#333FB1}{3}}\times \color{#333FB1}{3}\times \cancel{\color{#3A9A38}{a}}\times \cancel{\color{#3A9A38}{a}}\times \cancel{\color{#3A9A38}{a}}\times \color{#3A9A38}{a}\times \cancel{\color{#EC0000}{b}}\times \color{#EC0000}{b}\times \color{#EC0000}{b}\times \cancel{\color{#FA7921}{c}}}{\cancel{\color{#333FB1}{2}}\times \cancel{\color{#333FB1}{3}}\times \cancel{\color{#3A9A38}{a}}\times \cancel{\color{#3A9A38}{a}}\times \cancel{\color{#3A9A38}{a}}\times \cancel{\color{#EC0000}{b}}\times \cancel{\color{#FA7921}{c}}\times \color{#FA7921}{c}}\\&=\dfrac{\color{#333FB1}{3}\times \color{#3A9A38}{a} \times \color{#EC0000}{b}\times \color{#EC0000}{b}}{\color{#FA7921}{c}} \\&= \dfrac{3ab^2}{c} \end{align}||

The fraction |\dfrac{18a^4b^3c}{6a^3b}| , when simplified, is |\dfrac{3\times a \times b\times b}{c}| or |\dfrac{3ab^2}{c}.|

When simplifying a fraction with more than one term in the numerator or denominator, other factorization methods must be used. This is a case of simplifying rational expressions.