Factoring Monomials

Factoring monomials is a way of breaking the monomial into smaller terms and expressing them in the product form.

For example, the monomial 6m³ is factored as:

• (2)(3m³) = 6m³
• (3)(2m³) = 6m³
• (2m)(3m²) = 6m³
• (2m²)(3m) = 6m³
• (6)(m³) = 6m³
• (6m)(m²) = 6m³
• (6m²)(m) = 6m³
• (6m³)(1) = 6m³

Also, we observe that the product of each of the factors gives back the original monomial 6m³. Thus, the possible factors of the monomial 6m³ are 1, 2, 3, 6, m, m², m³, 2m, 2m², 2m³, 3m, 3m², 3m³, 6m, 6m², 6m³.

However, finding factors with this method takes time. Instead, we expand the monomials completely by factoring like prime factorization of integers. 

Factoring Completely

While factoring a monomial completely, we expand each part of the monomial until it is no longer expandable.

Again, taking the monomial 6m³ and completely factoring it, we get

6m³ = 2 × 3 × m × m × m

Thus, 2 × 3 × m × m × m is the complete factorization of 6m³.

What are the factors of 35v?

Solution:

35v = (1)(35v), (5)(7v), (7)(5v), (35)(v).
Thus, the factors of 35v are 1, 5, 7, 35, v, 5v, 7v, 35v.

Factor the monomial 16x²y completely.

Solution:

16x²y = 2 × 2 × 2 × 2 × x × x × y
Thus, 2 × 2 × 2 × 2 × x × x × y is the complete factorization of 16x²y.

Greatest Common Factor of Two or More Monomials

The greatest common factor (GCF) of monomials, also called the common monomial factor, refers to the product of the greatest common factor of coefficients and the greatest common factor of the variables.

To find the GCF of two or more monomials, we factor them completely and then find the product of all the common factors.

Now, let us find the GCF of 21p⁴q² and -14q⁵p³.

Factoring them completely, we get

21p⁴q² = 3 × 7 × p × p × p × p × q × q

-14q⁵p³ = -1 × 2 × 7 × q × q × q × q × q × p × p × p

Thus, the GCF is 7 × p × p × p × q × q.

Find the GCF of 36m²n², 20mn, and 40mn².

Solution:

Factoring the given monomials completely, we get
36m²n² = 2 × 2 × 3 × 3 × m × m × n × n
20mn = 2 × 2 × 5 × m × n
40mn² = 2 × 2 × 2 × 5 × m × n × n
Thus, the GCF of 36m²n², 20mn, and 40mn² = 2 × 2 × m × n = 4mn.