Prime Factorization of Monomials  Questions With Detailed Solutions
What is prime factorization of monomials? Grade 11 examples and questions are presented along with Solutions and explanations included.
Review of Monomials and Prime Factorization
A monomial is the product of a real number and variables raised to non negative integer powers.
Example of monomials: 2 x , 5 x^{ 2}y , 30 x^{ 2}y^{ 4}
What is prime factorization of monomials?
The prime factorization of a monomials is obtained by writing the whole number in the monomial in prime factorization followed by the product of the variables.
Examples of monomials in prime factorization form
1) 10 x = 2 × 5 × x
2) 20 x^{ 2} = 2 × 2 × 5 × x × x
3)  30 x^{ 2}y^{ 3} =  2 × 3 × 5 × x × x × y × y × y
Questions

Which of the following is not a prime factorization?
a)2 × 10 × x
b) 2 × 7 × x × x
c) 4 × 4 × 4 × x × x × y × y × y
d) 2 × 2 × 2 × 3 × x × x × y × y

What is the prime factorization of the following monomials?
a) 28 x y^{ 2}
b) 32 x ^{ 3}y
c) 70 x ^{ 3}y ^{ 3}
d) 120 x ^{ 2}y ^{ 2}

Find the prime factorizations of 5 x y^{ 2} and 20 x^{ 3}y and then the prime factorization of 100 x^{ 4}y^{ 3} knowing that 100 x^{ 4}y^{ 3} = (5 x y^{ 2}) × (20 x^{ 3}y)
Solutions to the Above Questions

Solution
a) The 10 in 2 × 10 × x is not a prime number and therefore 2 × 10 × x is not a prime factorization.
b) 2 × 7 × x × x is a prime factorization.
c) The 4 in 4 × 4 × 4 × x × x × y is not a prime number and therefore 4 × 4 × 4 × x × x × y is not a prime factorization.
d) 2 × 2 × 2 × 3 × x × x × y × y is a prime factorization.

Solution
a) 28 x y^{ 2} = 2 × 2 × 7 × x × y × y
b) 32 x ^{ 3}y = 2 × 2 × 2 × 2 × 2 × x × x × x × y
c) 70 x ^{ 3}y ^{ 3} = 2 ×5 × 7 × x × x × x × y × y × y
d) 120 x ^{ 2}y ^{ 2} = 2 × 2 × 2 × 3 × 5 × x× x × y

Solution
The prime factorization of 5 x y^{ 2} is.
5 x y^{ 2} = 5 × x × y × y
The prime factorization of 20 x^{ 3}y is.
20 x^{ 3}y = 2 × 2 × 5 × x × x × x × y
The prime factorization of 100 x^{ 4}y^{ 3} is.
100 x^{ 4}y^{ 3} = (5 x y^{ 2}) × (20 x^{ 3}y) =
(5 × x × y × y ) × (2 × 2 × 5 × x × x × x × y ) =
2 × 2 × 5 × 5 x × x × x × x × y × y × y

More References and Links
Prime Factorization
Prime Factors
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