Factoring consists of writing an expression in the form of a product of prime factors, called prime factorization. When factoring a monomial, its coefficient and variables must be decomposed.
To factor a monomial, follow these steps.
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Decompose the coefficient into prime factors.
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Decompose the variables.
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Write the monomial as a product of prime factors.
Factor the monomial |300x^3yz^2.|
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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|| -
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}|| -
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}.|
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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}|| -
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}|| -
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}}|| -
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.