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 Ã— 3 Ã— m Ã— m Ã— m

Thus, 2 Ã— 3 Ã— m Ã— m Ã— 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 Ã— 2 Ã— 2 Ã— 2 Ã— x Ã— x Ã— y
Thus, 2 Ã— 2 Ã— 2 Ã— 2 Ã— x Ã— x Ã— 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 Ã— 7 Ã— p Ã— p Ã— p Ã— p Ã— q Ã— q

-14q5p3 = -1 Ã— 2 Ã— 7 Ã— q Ã— q Ã— q Ã— q Ã— q Ã— p Ã— p Ã— p

Thus, the GCF is 7 Ã— p Ã— p Ã— p Ã— q Ã— q.

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

Solution:

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